Method of analyzing a subterranean formation and method of producing a mineral hydrocarbon fluid from the formation

ABSTRACT

Method of analyzing a subterranean formation traversed by a wellbore. The method uses a tool comprising a transmitter antenna and a receiver antenna, the subterranean formation comprising one or more formation layers. The tool is suspended inside the wellbore, and one or more electromagnetic fields are induced in the formation. One or more time-dependent transient response signals are detected and analyzed. Electromagnetic anisotropy of at least one of the formation layers is detectable. Geosteering cues may be derived from the time-dependent transient response signals, for continued drilling of the well bore until a hydrocarbon reservoir is reached. The hydrocarbon may then be produced.

CROSS REFERENCE TO EARLIER APPLICATION

The present application claims benefit under 35 USC § 119(e) of U.S.Provisional application No. 60/797,556 filed 4 May 2006.

FIELD OF THE INVENTION

In one aspect, the present invention relates to a method of analyzing asubterranean formation traversed by a wellbore. In another aspect theinvention relates to a method of producing a mineral hydrocarbon fluidfrom an earth formation. In still another aspect, the invention relatesto a computer readable medium storing computer readable instructionsthat analyze one or more electromagnetic response signals.

BACKGROUND OF THE INVENTION

In logging while drilling (LWD) geo-steering applications, it isadvantageous to detect the presence of a formation anomaly ahead of oraround a bit or bottom hole assembly. There are many instances where“Look-Ahead” capability is desired in LWD logging environments.Look-ahead logging comprises detecting an anomaly at a distance ahead ofa drill bit. Some look-ahead examples include predicting anover-pressured zone in advance, or detecting a fault in front of thedrill bit in horizontal wells, or profiling a massive salt structureahead of the drill bit.

In U.S. Pat. No. 5,955,884 to Payton, et al, a tool and method aredisclosed for transient electromagnetic logging, wherein electric andelectromagnetic transmitters are utilized to apply electromagneticenergy to a formation at selected frequencies and waveforms thatmaximize radial depth of penetration into the target formation. In thistransient EM method, the current applied at a transmitter antenna isgenerally terminated and a temporal change of voltage induced in areceiver antenna is monitored over time.

When logging measurements are used for well placement, detection oridentification of anomalies can be critical. Such anomalies may includefor example, a fault, a bypassed reservoir, a salt dome, or an adjacentbed or oil-water contact.

U.S. patent applications published under Nos. 2005/0092487,2005/0093546, 2006/0038571, each incorporated herein by reference,describe methods for localizing such anomalies in a subterranean earthformation employing transient electromagnetic (EM) reading. The methodsparticularly enable finding the direction and distance to a resistive orconductive anomaly in a formation surrounding a borehole, or ahead ofthe borehole, in drilling applications.

Of the referenced U.S. patent application publications, US 2006/0038571shows that transient electromagnetic responses can be analyzed todetermine conductivity values of a homogeneous earth formation (singlelayer), and of two or three or more earth layers, as well as distancesfrom the tool to the interfaces between the earth layers.

In principle, the methodology as set forth in US 2006/0038571 would workfor any number of layers. However, the larger the number of layers, andparticularly when the layers are thin, the more complicated the analysisis. For instance, a thinly laminated sand/shale sequence would bedifficult to analyze employing the methodology as set forth in US2006/0038571.

SUMMARY OF THE INVENTION

In accordance with the invention there is provided a method of analyzinga subterranean formation traversed by a wellbore, using a toolcomprising a transmitter antenna and a receiver antenna, thesubterranean formation comprising one or more formation layers and themethod comprising:

suspending the tool inside the wellbore;

inducing one or more electromagnetic fields in the formation;

detecting one or more time-dependent transient response signals;

analyzing the one or more time-dependent transient response signalstaking into account electromagnetic anisotropy of at least one of theformation layers.

The electromagnetic properties of a formation layer comprising a numberof thin layers may be approximated by one formation layer comprising anelectromagnetic anisotropy. It is thereby avoided to have to take intoaccount each thin layer individually when inverting the responses.

Amongst other advantages of taking into account electromagneticanisotropy, is that anisotropy information may be useful in preciselylocating mineral hydrocarbon fluid containing reservoirs, as suchreservoirs are often associated with electromagnetic anisotropy offormation layers.

The result of the analyzing step mentioned above may be outputted,including displayed or stored or transmitted or otherwise made conveyedand made available to an operator or a geosteering system. Such ageosteering system may use the result of the analysis to generate ageosteering cue in response. The geosteering cue may in itself beoutputted, including displayed or stored or transmitted or otherwisemade conveyed and made available to an operator and/or used to continuedrilling in response to the geosteering cue.

Said method of analyzing a subterranean formation may be used in ageosteering application, wherein a geosteering cue may be derived fromthe one or more time-dependent transient response signals, taking intoaccount electromagnetic anisotropy, and wherein a drilling operation maybe continued in accordance with the derived geosteering cue in order toaccurately place a well.

Thus, in another aspect there is provided a method of producing amineral hydrocarbon fluid from an earth formation, the method comprisingsteps of:

suspending a drill string in the earth formation, the drill stringcomprising at least a drill bit and measurement sub comprising atransmitter antenna and a receiver antenna;

drilling a well bore in the earth formation;

inducing an electromagnetic field in the earth formation employing thetransmitter antenna;

detecting a transient electromagnetic response signal from theelectromagnetic field, employing the receiver antenna;

deriving a geosteering cue from the electromagnetic response;

continue drilling the well bore in accordance with the geosteering cueuntil a reservoir containing the hydrocarbon fluid is reached;

producing the hydrocarbon fluid.

In still another aspect, the invention provides a computer readablemedium storing computer readable instructions that analyze one or moredetected time-dependent transient electromagnetic response signals thathave been detected by a tool suspended inside a wellbore traversing asubterranean formation after inducing one or more electromagnetic fieldsin the formation, wherein the computer readable instructions take intoaccount electromagnetic anisotropy of at least one formation layer inthe subterranean formation.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention is described in more detail below by way ofexamples and with reference to the attached drawing figures, wherein:

FIG. 1A is a block diagram showing a system implementing embodiments ofthe invention;

FIG. 1B schematically illustrates an alternative system implementingembodiments of the invention;

FIG. 2 is a flow chart illustrating a method in accordance with anembodiment of the invention;

FIG. 3 is a graph illustrating directional angles between toolcoordinates and anomaly coordinates;

FIG. 4A is a graph showing a resistivity anomaly in a tool coordinatesystem;

FIG. 4B is a graph showing a resistivity anomaly in an anomalycoordinate system;

FIG. 5 is a graph illustrating tool rotation within a borehole;

FIG. 6 schematically shows directional components involvingelectromagnetic induction tools relative to an electromagnetic inductionanomaly;

FIG. 7 is a graph showing the voltage response from coaxial V_(zz)(t),coplanar V_(xx)(t), and the cross-component V_(zx)(t) measurements forL=1 m, for θ=30°, and a distance D=10 m from a salt layer;

FIG. 8 is a graph showing the voltage response from coaxial V_(zz)(t),coplanar V_(xx)(t), and the cross-component V_(zx)(t) measurements forL=1 m, for θ=30°, and a distance D=100 m from a salt layer;

FIG. 9 is a graph showing apparent dip (θ_(app)(t)) for an arrangementas in FIG. 7;

FIG. 10 is a graph showing apparent conductivity (σ_(app)(t)) calculatedfrom both the coaxial (V_(zz)(t)) and the coplanar (V_(xx)(t)) responsesfor the same conditions as in FIG. 9;

FIG. 11 is a graph showing apparent dip θ_(app)(t) for the L=1 m toolassembly when the salt face is D=10 m away, for various angles betweenthe tool axis and the target;

FIG. 12 is a graph similar to FIG. 11 whereby the salt face is D=50 maway from the tool;

FIG. 13 is a graph similar to FIG. 11 whereby the salt face is D=100 maway from the tool;

FIG. 14 is a schematic illustration showing a coaxial tool with its toolaxis parallel to a layer interface;

FIG. 15 is a graph showing transient voltage response as a function of tas given by the coaxial tool of FIG. 14 in a two-layer formation atdifferent distances from the bed;

FIG. 16 is a graph showing the voltage response data of FIG. 15 in termsof the apparent conductivity (σ_(app)(t));

FIG. 17 is similar to FIG. 16 except that the resistivities of layers 1and 2 have been interchanged;

FIG. 18 shows a graph of the σ_(app)(t) for the case D=1 m and L=1 m,for various resistivity ratios while the target resistivity is fixed atR₂=1 Ωm;

FIG. 19 shows a comparison of apparent conductivity at large values oft, σ_(app)(t→∞), for coaxial responses where D=1 m and L=1 m as afunction of conductivity σ₂ of the target layer while the localconductivity σ₁ is fixed at 1 S/m;

FIG. 20 graphically shows the same data as FIG. 19 plotted as the ratioof target conductivity over local layer conductivity σ₁ versus ratio ofthe late time apparent conductivity σ_(app)(t→∞) over local layerconductivity σ₁;

FIG. 21 shows a graph containing apparent conductivity (σ_(app)(t))versus time for various combinations of D and L;

FIG. 22 graphically shows the relationship between ray-path RP andtransition time t_(c);

FIG. 23 is a schematic illustration showing a coaxial tool approachingor just beyond a bed boundary;

FIG. 24 is a graph showing transient voltage response as a function of tas given by the coaxial tool of FIG. 23 at different distances D fromthe bed;

FIG. 25 is a graph showing the voltage response data of FIG. 24 in termsof the apparent conductivity (σ_(app)(t));

FIG. 26 is similar to FIG. 25 except that the resistivities of layers 1and 2 have been interchanged;

FIG. 27 presents a graph comparing σ_(app)(t) of FIG. 25 and FIG. 26relating to D=1 m;

FIG. 28 shows a graph of σ_(app)(t) on a linear scale for varioustransmitter/receiver spacings L in case D=50 m;

FIG. 29 graphically shows distance to anomaly ahead of the tool versestransition time (t_(c)) as determined from the data of FIG. 25;

FIG. 30 schematically shows a coplanar tool approaching or just beyond abed boundary;

FIG. 31 is a graph showing transient voltage response data in terms ofthe apparent conductivity (σ_(app)(t)) as a function of t as provided bythe coplanar tool of FIG. 30 at different distances D from the bed;

FIG. 32 shows a comparison of the late time apparent conductivity(σ_(app)(t→∞)) for coplanar responses where D=50 m and L=1 m as afunction of conductivity σ₁ of the local layer while the targetconductivity σ₂ is fixed at 1 S/m;

FIG. 33 graphically shows the same data as FIG. 32 plotted as the ratioof target conductivity σ₂ over local layer conductivity σ₁ versus ratioof the late time apparent conductivity σ_(app)(t→∞) over local layerconductivity σ₁;

FIG. 34 graphically shows distance to anomaly ahead of the tool versestransition time (t_(c)) as determined from the data of FIG. 31;

FIG. 35 schematically shows a model of a coaxial tool in a conductivelocal layer (1 Ωm), a very resistive layer (100 Ωm), and a furtherconductive layer (1 Ωm);

FIG. 36 is a graph showing apparent resistivity response versus time,R_(app)(t), for a geometry as given in FIG. 35 for various thicknesses Δof the very resistive layer;

FIG. 37 schematically shows a model of a coaxial tool in a resistivelocal layer (10 Ωm), a conductive layer (1 Ωm), and a further resistivelayer (10 Ωm);

FIG. 38 is a graph similar to FIG. 36, showing apparent resistivityresponse R_(app)(t) versus time for a geometry as given in FIG. 37 forvarious thicknesses Δ of the conductive layer;

FIG. 39 schematically shows a model of a coaxial tool in a conductivelocal layer (1 Ωm) in the vicinity of a highly resistive layer (100 Ωm)with a separating layer having an intermediate resistance (10 Ωm) ofvarying thickness in between;

FIG. 40 is a graph similar to FIG. 36, showing apparent resistivityresponse versus time, R_(app)(t), for a geometry as given in FIG. 39 forvarious thicknesses Δ of the separating layer;

FIG. 41 shows calculated coaxial transient voltage responses for an L=1m tool in an anisotropic formation wherein σ_(H)=1 S/m (R_(H)=1 Ωm) forvarious values of β²;

FIG. 42 shows apparent conductivity based on the responses of FIG. 41;

FIG. 43 shows apparent conductivity based on coaxial responses for anL=1 m tool in a formation wherein σ_(H)=0.1 S/m for various values ofβ²;

FIG. 44 shows apparent conductivity based on coaxial responses for anL=1 m tool in a formation wherein σ_(H)=0.01 S/m for various values ofβ²;

FIG. 45 shows a graph plotting late time asymptotic value of coaxialapparent conductivity σ_(Zz)(t→∞) from FIGS. 44 to 44, normalized byσ_(H), against a variable representing β²;

FIG. 46 shows apparent dip angle θ_(app)(t) as a function of time basedon calculated coaxial, coplanar and cross-component transient responsesfrom a L=1 m tool in a formation of R_(H)=10 Ωm and R_(V)/R_(H)=9;

FIG. 47 shows an electromagnetic induction tool in a formation layercomprising a package of alternating sets of sub-layers;

FIG. 48 shows a graph of apparent resistivity in co-axial measurementand co-planar measurement of the geometry as in FIG. 47;

FIG. 49 schematically shows directional components of an electromagneticinduction tool relative to an anisotropic anomaly;

FIG. 50 shows a plot of the apparent conductivity (σ_(app)(z; t)) inboth z- and t-coordinates for various distances D;

FIG. 51 shows a plot of the apparent conductivity (σ_(app)(z; t)) inboth z- and t-coordinates;

FIG. 52 schematically shows a model of a structure involving a highlyresistive layer (100 Ωm) covered by a conductive local layer (1 Ωm)which is covered by a resistive layer (10 Ωm), whereby a coaxial tool isdepicted in the resistive layer;

FIG. 53A shows apparent resistivity in both z and t coordinates wherebyinflection points are joined using curve fitted lines;

FIG. 53B shows an image log derived from FIG. 53A;

FIG. 54A schematically shows a coaxial tool seen as approaching a highlyresistive formation at a dip angle of approximately 30 degrees;

FIG. 54B shows apparent dip response in both t and z coordinates forz-locations corresponding to those depicted in FIG. 54A.

DETAILED DESCRIPTION OF THE INVENTION

The present invention will now be described in relation to particularembodiments, which are intended in all respects to be illustrativerather than restrictive. Alternative embodiments will become apparent tothose skilled in the art to which the present invention pertains withoutdeparting from its scope.

It will be understood that certain features and sub-combinations are ofutility and may be employed without reference to other features andsub-combinations specifically set forth. This is contemplated and withinthe scope of the claims.

Embodiments of the invention relate to analysis of electromagnetic (EM)induction signals and to a system and method for determining distanceand/or direction to an anomaly in a formation from a location within awellbore. The analysis is sensitive to electromagnetic anomalies, inparticular electromagnetic induction anomalies.

Both frequency domain excitation and time domain excitation have beenused to excite electromagnetic fields for use in anomaly detection. Infrequency domain excitation, a device transmits a continuous wave of afixed or mixed frequency and measures responses at the same band offrequencies. In time domain excitation, a device transmits a square wavesignal, triangular wave signal, pulsed signal or pseudo-random binarysequence as a source and measures the broadband earth response. Suddenchanges in transmitter current cause transient signals to appear at areceiver caused by induction currents in the formation. The signals thatappear at the receiver are called transient responses because thereceiver signals start at a first value after a sudden change intransmitter current, and then they decay (or increase) with time to anew constant level at a second value. The technique disclosed hereinimplements the time domain excitation technique.

As set forth below, embodiments of the invention propose a generalmethod to determine a direction from a measurement sub to a resistive orconductive anomaly using transient EM responses. As will be explained indetail, the direction to the anomaly is specified by a dip angle and anazimuth angle. Embodiments of the invention propose to define anapparent dip (θ_(app)(t)) and an apparent azimuth (φ_(app)(t)) bycombinations of multi-axial, e.g. bi-axial or tri-axial, transientmeasurements. The true direction, in terms of dip and azimuth angles({θ, φ}), may be determined from the analysis of the apparent direction({θ_(app)(t), φ_(app)(t)}). For instance, the apparent direction({θ_(app)(t), φ_(app)(t)}) approaches the true direction ({θ, φ}) as atime (t) increases, if the anomaly has a high thickness as seen from thetool.

Time-dependent values for apparent conductivity may be obtained fromcoaxial and coplanar electromagnetic induction measurements, and canrespectively be denoted as σ_(coaxial)(t) and σ_(coplanar)(t). Both readthe conductivity in the total present formation around the tool. Theθ_(app)(t) and φ_(app)(t) both initially read zero when an apparentconductivity σ_(coaxial)(t) and σ_(coplanar)(t) from coaxial andcoplanar measurements both read the conductivity of the formationsurrounding the tool nearby. The apparent conductivity will be furtherexplained below and can also be used to determine the location of ananomaly in a wellbore.

Whenever in the present specification the term “conductivity” isemployed, it is intended to cover also its inverse equivalent“resistivity”, and vice versa. The same holds for the terms “apparentconductivity” and “apparent resistivity”.

FIGS. 1A and 1B illustrate systems that may be used to implement theembodiments of the method of the invention. A surface computing unit 10may be connected with an electromagnetic measurement tool 2 disposed ina wellbore 4.

In FIG. 1A, the tool 2 is suspended on a cable 12. The cable 12 may beconstructed of any known type of cable for transmitting electricalsignals between the tool 2 and the surface computing unit 10.

In FIG. 1B, the tool is comprised in a measurement sub 11 and suspendedin the wellbore 4 by a drill string 15. The drill string 15 furthersupports a drill bit 17, and may support a steering system 19. Thesteering system may be of a known type, including a rotatable steeringsystem or a sliding steering system. The wellbore 4 traverses the earthformation 5 and it is an objective to precisely direct the drill bit 17into a hydrocarbon fluid containing reservoir 6 to enable producing thehydrocarbon fluid via the wellbore. Such a reservoir 6 may manifestitself as an electromagnetic anomaly in the formation 5.

Referring again to both FIGS. 1A and 1B, one or more transmitters 16 andone are more receivers 18 may be provided for transmitting and receivingelectromagnetic signals into and from the formation around the wellbore4. A data acquisition unit 14 may be provided to transmit data to andfrom the transmitters 16 and receivers 18 to the surface computing unit10.

Each transmitter 16 and/or receiver 18 may comprise a coil, wound arounda support structure such as a mandrel. The support structure maycomprise a non-conductive section to suppress generation of eddycurrents. The non-conductive section may comprise one or more slots,optionally filled with a non-conductive material, or it may be formedout of a non-conductive material such as a composite plastic.Alternatively, the support structure is coated with a layer of ahigh-magnetic permeable material to form a magnetic shield between theantenna and the support structure.

Each transmitter 16 and each receiver 18 may be bi-axial or eventri-axial, and thereby contain components for sending and receivingsignals along each of three axes. Accordingly, each transmitter modulemay contain at least one single or multi-axis antenna and may be a3-orthogonal component transmitter. Each receiver may include at leastone single or multi-axis electromagnetic receiving component and may bea 3-orthogonal component receiver.

A tool/borehole coordinate system is defined as having x, y, and z axes.The z-axis defines the direction from the transmitter T to the receiverR. It will be assumed hereinafter that the axial direction of thewellbore 4 coincides with the z-axis, whereby the x- and y-axescorrespond to two orthogonal directions in a plane normal to thedirection from the transmitter T to the receiver R and to the wellbore4.

The data acquisition unit 14 may include a controller for controllingthe operation of the tool 2. The data acquisition unit 14 preferablycollects data from each transmitter 16 and receiver 18 and provides thedata to the surface computing unit 10. The data acquisition unit 14 maycomprise an amplifier and/or a digital to analogue converter, asdescribed in co-pending U.S. application Ser. No. 11/689,980 filed on 22Mar. 2007, incorporated herein by reference, to amplify the responsesand/or convert to a digital representation of the responses beforetransmitting to the surface computing unit 10 via cable 12 and/or anoptional telemetry unit 13.

The surface computing unit 10 may include computer components includinga processing unit 30, an operator interface 32, and a tool interface 34.The surface computing unit 10 may also include a memory 40 includingrelevant coordinate system transformation data and assumptions 42, anoptional direction calculation module 44, an optional apparent directioncalculation module 46, and an optional distance calculation module 48.The optional direction and apparent direction calculation modules aredescribed in more detail in already incorporated US patent applicationpublication 2005/0092487 and need not be further described here, otherthan specifying that these optional modules may take into accountformation anisotropy.

The surface computing unit 10 may include computer components includinga processing unit 30, an operator interface 32, and a tool interface 34.The surface computing unit 10 may also include a memory 40 includingrelevant coordinate system transformation data and assumptions 42, adirection calculation module 44, an apparent direction calculationmodule 46, and a distance calculation module 48. The surface computingunit 10 may further include a bus 50 that couples various systemcomponents including the system memory 40 to the processing unit 30. Thecomputing system environment 10 is only one example of a suitablecomputing environment and is not intended to suggest any limitation asto the scope of use or functionality of the invention. Furthermore,although the computing system 10 is described as a computing unitlocated on a surface, it may optionally be located below the surface,incorporated in the tool, positioned at a remote location, or positionedat any other convenient location.

The memory 40 preferably stores one or more of modules 48, 44 and 46,which may be described as program modules containing computer-executableinstructions, executable by the surface computing unit 10. Each modulemay comprise or make use of a computer readable medium that storescomputer readable instructions for analyzing one or more detectedtime-dependent transient electromagnetic response signals that have beendetected by a tool suspended inside a wellbore traversing a subterraneanformation after inducing one or more electromagnetic fields in theformation. The instructions may implement any part of the disclosurethat follows herein below.

For example, the program module 44 may contain computer executableinstructions to calculate a direction to an anomaly within a wellbore.The program module 48 may contain computer executable instructions tocalculate a distance to an anomaly or a thickness of the anomaly. Thestored data 42 may include data pertaining to the tool coordinate systemand the anomaly coordinate system and other data for use by the programmodules 44, 46, and 48. Preferably, the computer readable instructionstake into account electromagnetic anisotropy of at least one formationlayer in the subterranean formation. For further details on thecomputing system 10, including storage media and input/output devices,reference is made to US patent application publication 2005/0092487,incorporated herein by reference. Accordingly, additional detailsconcerning the internal construction of the computer 10 need not bedisclosed in connection with the present invention.

FIG. 2 is a flow chart illustrating the procedures involved in a methodembodying the invention. Generally, in procedure A, the transmitters 16transmit electromagnetic signals. In procedure B, the receivers 18receive transient responses. In procedure C, the system processes thetransient responses. The procedures may then end or start again.

Procedure C may comprise determining a distance and/or a direction tothe anomaly may be determined. Procedure C may comprise creating animage of formation features based on the transient electromagneticresponses. Electromagnetic anisotropy of at least one of the formationlayers may be taken into account.

FIGS. 3-6 illustrate the technique for implementing procedure C fordetermining distance and/or direction to the anomaly. FIGS. 6 and 41 to49 illustrate how electromagnetic anisotropy may be taken into account,e.g. in determining distance and/or direction to the anomaly.

Tri-Axial Transient EM Responses

FIG. 3 illustrates directional angles between tool coordinates andanomaly coordinates. A transmitter coil T is located at an origin thatserves as the origin for each coordinate system. A receiver R is placedat a distance L from the transmitter. An earth coordinate system,includes a Z-axis in a vertical direction and an X-axis and a Y-axis inthe East and the North directions, respectively. The deviated boreholeis specified in the earth coordinates by a deviation angle θ_(b) and itsazimuth angle φ_(b). A resistivity anomaly A is located at a distance Dfrom the transmitter in the direction specified by a dip angle (θ_(a))and its azimuth (φ_(a)).

In order to practice embodiments of the method, FIG. 4A shows thedefinition of a tool/borehole coordinate system having x, y, and z axes.The z-axis defines the direction from the transmitter T to the receiverR. The tool coordinates in FIG. 4A are specified by rotating the earthcoordinates (X, Y, Z) in FIG. 3 by the azimuth angle (φ_(b)) around theZ-axis and then rotating by θ_(b) around the y-axis to arrive at thetool coordinates (x, y, z). The direction of the anomaly is specified bythe dip angle (ν) and the azimuth angle (φ) where:

$\begin{matrix}{{\cos \; \vartheta} = {\left( {{\hat{b}}_{z} \cdot \hat{a}} \right) = {{\cos \; \theta_{a}\cos \; \theta_{b}} + {\sin \; \theta_{a}\sin \; \theta_{b}{\cos \left( {\phi_{a} - \phi_{b}} \right)}}}}} & (1) \\{{\tan \; \varphi} = \frac{\sin \; \theta_{b}{\sin \left( {\phi_{a} - \phi_{b}} \right)}}{{\cos \; \theta_{a}\sin \; \theta_{b}{\cos \left( {\phi_{a} - \phi_{b}} \right)}} - {\sin \; \theta_{a}\cos \; \theta_{b}}}} & (2)\end{matrix}$

Similarly, FIG. 4B shows the definition of an anomaly coordinate systemhaving a, b, and c axes. The c-axis defines the direction from thetransmitter T to the center of the anomaly A. The anomaly coordinates inFIG. 4B are specified by rotating the earth coordinates (X, Y, Z) inFIG. 3 by the azimuth angle (φ_(a)) around the Z-axis and subsequentlyrotating by θ_(a) around the b-axis to arrive at the anomaly coordinates(a, b, c). In this coordinate system, the direction of the borehole isspecified in a reverse order by the azimuth angle (φ) and the dip angle(ν).

Transient Responses in Two Coordinate Systems

The method is additionally based on the relationship between thetransient responses in two coordinate systems. The magnetic fieldtransient responses at the receivers [R_(x), R_(y), R_(z),] which areoriented in the [x, y, z] axis direction of the tool coordinates,respectively, are noted as

$\begin{matrix}{{\begin{bmatrix}V_{xx} & V_{xy} & V_{xz} \\V_{yx} & V_{yy} & V_{yz} \\V_{zx} & V_{zy} & V_{zz}\end{bmatrix} = {\begin{bmatrix}R_{x} \\R_{y} \\R_{z}\end{bmatrix}\begin{bmatrix}T_{x} & T_{y} & T_{z}\end{bmatrix}}},} & (3)\end{matrix}$

wherein the right-hand side of the equation represents all combinationsof receiver axis and transmitter axis, whereby V_(ij)=R_(i)T_(j) denotesvoltage response sensed by receiver R_(i) (i=x, y, z) from signaltransmitted by transmitter T_(j) (j=x, y, z). Each transmitter maycomprise a magnetic dipole source, [M_(x), M_(y), M_(z)], in anydirection.

When the resistivity anomaly is distant from the tool, the formationnear the tool is seen as a homogeneous formation. For simplicity, themethod may assume that the formation is isotropic. Only three non-zerotransient responses exist in a homogeneous isotropic formation. Theseinclude the coaxial response and two coplanar responses. Coaxialresponse V_(zz)(t) is the response when both the transmitter and thereceiver are oriented in the common tool axis direction. Coplanarresponses, V_(xx)(t) and V_(yy)(t), are the responses when both thetransmitter T and the receiver R are aligned parallel to each other buttheir orientation is perpendicular to the tool axis. All of thecross-component responses are identically zero in a homogeneousisotropic formation. Cross-component responses are either from alongitudinally oriented receiver with a transverse transmitter, or viseversa. Another cross-component response is also zero between a mutuallyorthogonal transverse receiver and transverse transmitter.

The effect of the resistivity anomaly is seen in the transient responsesas time increases. In addition to the coaxial and the coplanarresponses, the cross-component responses V_(ij)(t) (i≠j; i, j=x, y, z)become non-zero.

The magnetic field transient responses may also be examined in theanomaly coordinate system. The magnetic field transient responses at thereceivers [R_(a), R_(b), R_(c),] that are oriented in the [a, b, c] axisdirection of the anomaly coordinates, respectively, may be noted as

$\begin{matrix}{\begin{bmatrix}V_{aa} & V_{ab} & V_{ac} \\V_{ba} & V_{bb} & V_{bc} \\V_{ca} & V_{cb} & V_{cc}\end{bmatrix} = {\begin{bmatrix}R_{a} \\R_{b} \\R_{c}\end{bmatrix}\begin{bmatrix}T_{a} & T_{b} & T_{c}\end{bmatrix}}} & (4)\end{matrix}$

wherein the right-hand side of the equation represents all combinationsof receiver orientation and transmitter orientation, wherebyV_(ij)=R_(i)T_(j) denotes voltage response sensed by receiver R_(i) (inorientation i=a, b, c) from signal transmitted by transmitter T_(j) (inorientation j=a, b, c). Each transmitter may comprise a magnetic dipolesource, [M_(a), M_(b), M_(c)], along the orientation a, b, or c.

When the anomaly is large and distant compared to thetransmitter-receiver spacing, the effect of spacing can be ignored andthe transient responses can be approximated with those of the receiversnear the transmitter. Then, the method assumes that axial symmetryexists with respect to the c-axis that is the direction from thetransmitter to the center of the anomaly. In such an axially symmetricconfiguration, the cross-component responses in the anomaly coordinatesare identically zero in time-domain measurements.

$\begin{matrix}{\begin{bmatrix}V_{aa} & V_{ab} & V_{ac} \\V_{ba} & V_{bb} & V_{bc} \\V_{ca} & V_{cb} & V_{cc}\end{bmatrix} = \begin{bmatrix}V_{aa} & 0 & 0 \\0 & V_{aa} & 0 \\0 & 0 & V_{cc}\end{bmatrix}} & (5)\end{matrix}$

The magnetic field transient responses in the tool coordinates arerelated to those in the anomaly coordinates by a simple coordinatetransformation P(ν, φ) specified by the dip angle (ν) and azimuth angle(φ).

$\begin{matrix}{\begin{bmatrix}V_{xx} & V_{xy} & V_{xz} \\V_{yx} & V_{yy} & V_{yz} \\V_{zx} & V_{zy} & V_{zz}\end{bmatrix} = {{{P\left( {\vartheta,\varphi} \right)}^{tr}\begin{bmatrix}V_{aa} & V_{ab} & V_{ac} \\V_{ba} & V_{bb} & V_{bc} \\V_{ca} & V_{cb} & V_{cc}\end{bmatrix}}{P\left( {\vartheta,\varphi} \right)}}} & (6) \\{{P\left( {\vartheta,\varphi} \right)} = \begin{bmatrix}{\cos \; {\vartheta cos}\; \varphi} & {\cos \; {\vartheta sin}\; \varphi} & {{- \sin}\; \vartheta} \\{{- \sin}\; \varphi} & {\cos \; \varphi} & 0 \\{\sin \; {\vartheta cos}\; \varphi} & {\sin \; {\vartheta sin}\; \varphi} & {\cos \; \vartheta}\end{bmatrix}} & (7)\end{matrix}$

Determination of Direction

The assumptions set forth above contribute to determination of targetdirection, which is defined as the direction of the anomaly from theorigin. The tool is in the origin. When axial symmetry in the anomalycoordinates is assumed, the transient response measurements in the toolcoordinates are constrained and the two directional angles may bedetermined by combinations of tri-axial responses.

$\begin{matrix}{\begin{bmatrix}V_{xx} & V_{xy} & V_{xz} \\V_{yx} & V_{yy} & V_{yz} \\V_{zx} & V_{zy} & V_{zz}\end{bmatrix} = {{{P\left( {\vartheta,\varphi} \right)}^{tr}\begin{bmatrix}V_{aa} & 0 & 0 \\0 & V_{aa} & 0 \\0 & 0 & V_{CC}\end{bmatrix}}{P\left( {\vartheta,\varphi} \right)}}} & (8)\end{matrix}$

In terms of each tri-axial response

V _(xx)=(V _(aa) cos² ν+V _(cc) sin²ν)cos² φ+V _(aa) sin²φ

V _(yy)=(V _(aa) cos² ν+V _(cc) sin²ν)sin² φ+V _(aa) cos²φ

V _(zz) =V _(aa) sin² ν+V _(cc) cos²ν  (9)

V _(xy) =V _(yx)=−(V _(aa) −V _(cc))sin²ν cos φ sin φ

V _(zx) =V _(xz)=−(V _(aa) −V _(cc))cos ν sin ν cos φ

V _(yz) =V _(zy)=−(V _(aa) −V _(cc))cos ν sin ν sin φ  (10)

The following relations can be noted:

V _(xx) +V _(yy) +V _(zz)=2V _(zz) +V _(cc)

V _(xx) −V _(yy)=(V _(cc) −V _(aa))sin²ν(cos²φ−sin²φ)

V _(yy) −V _(zz)=−(V _(cc) −V _(aa))(cos²ν−sin²ν sin²φ)

V _(zz) −V _(xx)=(V _(cc) −V _(aa))(cos²ν−sin²ν cos²φ)  (11)

Several distinct cases can be noted. In the first of these cases, whennone of the cross-components is zero, V_(xy)≠0 nor V_(yz)≠0 norV_(zx)≠0, then the azimuth angle φ is not zero nor π/2 (90°), and can bedetermined by,

$\begin{matrix}{\varphi = {\frac{1}{2}\tan^{- 1}\frac{V_{xy} + V_{yx}}{V_{xx} - V_{yy}}}} & (12)\end{matrix}$

By noting the relation,

$\begin{matrix}{\frac{V_{xy}}{V_{xz}} = {{\tan \; {\vartheta sin}\; \varphi \mspace{14mu} {and}\mspace{14mu} \frac{V_{xy}}{V_{yz}}} = {\tan \; {\vartheta cos}\; \varphi}}} & (13)\end{matrix}$

the dip (deviation) angle ν is determined by,

$\begin{matrix}{{\tan \; \vartheta} = \sqrt{\left( \frac{V_{xy}}{V_{xz}} \right)^{2} + \left( \frac{V_{xy}}{V_{yz}} \right)^{2}}} & (14)\end{matrix}$

In the second case, when V_(xy)=0 and V_(yz)=0, then ν=0 or φ=0 or π(180°) or φ=±π/2 (90°) and ν=±π/2 (90°), as the coaxial and the coplanarresponses should differ from each other (V_(aa)≠V_(cc)). If φ=0, thenthe dip angle ν is determined by,

$\begin{matrix}{\vartheta = {{- \frac{1}{2}}\tan^{- 1}\frac{V_{xz} + V_{zx}}{V_{xx} - V_{zz}}}} & (15)\end{matrix}$

If φ=π (180°), then the dip angle ν is determined by,

$\begin{matrix}{\vartheta = {{+ \frac{1}{2}}\tan^{- 1}\frac{V_{xz} + V_{zx}}{V_{xx} - V_{zz}}}} & (16)\end{matrix}$

Also, with regard to the second case, If ν=0, then V_(xx)=V_(yy) andV_(zx)=0. If φ=±π/2 (90°) and ν=±π/2 (90°), then V_(zz)=V_(xx) andV_(zx)=0. These instances are further discussed below with relation tothe fifth case.

In the third case, when V_(xy)=0 and V_(xz)=0, then φ=±π/2 (90°) or ν=0or φ=0 and ν=±π/2 (90°).

If φ=π/2, then the dip angle ν is determined by,

$\begin{matrix}{\vartheta = {{- \frac{1}{2}}\tan^{- 1}\frac{V_{yz} + V_{zy}}{V_{yy} - V_{zz}}}} & (17)\end{matrix}$

If φ=−π/2, then the dip angle ν is determined by,

$\begin{matrix}{\vartheta = {{+ \frac{1}{2}}\tan^{- 1}\frac{V_{yz} + V_{zy}}{V_{yy} - V_{zz}}}} & (18)\end{matrix}$

Also with regard to the third case, If ν=0, then V_(xx)=V_(yy) andV_(yz)=0. If φ=0 and ν=±π/2 (90°), V_(yy)=V_(zz) and V_(yz)=0. Thesesituations are further discussed below with relation to the fifth case.

In the fourth case, V_(xz)=0 and V_(yz)=0, then ν=0 or π(180°) or ±π/2(90°).

If ν=±π/2, then the azimuth angle φ is determined by,

$\begin{matrix}{\varphi = {{- \frac{1}{2}}\tan^{- 1}\frac{V_{xy} + V_{yx}}{V_{xx} - V_{yy}}}} & (19)\end{matrix}$

Also with regard to the fourth case, if ν=0 or π(180°), thenV_(xx)=V_(yy) and V_(yz)=0. This situation is also shown below withrelation to the fifth case.

In the fifth case, all cross components vanish, V_(xz)=V_(yz)=V_(xy)=0,then ν=0, or ν=±π/2 (90°) and φ=0 or ±π/2 (90°).

If V_(xx)=V_(yy) then ν=0 or π(180°).

If V_(yy)=V_(zz) then ν=±π/2 (90°) and φ=0.

If V_(zz)=V_(xx) then ν=±π/2 (90°) and φ=±π/2 (90°).

Tool Rotation Around the Tool/Borehole Axis

In the above analysis, all the transient responses V_(ij)(t) (i, j=x, y,z) are specified by the x-, y-, and z-axis directions of the toolcoordinates. However, the tool rotates inside the borehole and theazimuth orientation of the transmitter and the receiver no longercoincides with the x- or y-axis direction as shown in FIG. 5. If themeasured responses are {tilde over (V)}_(ĩ{tilde over (j)})(ĩ,{tildeover (j)}={tilde over (x)},{tilde over (y)},{tilde over (z)}), where{tilde over (x)} and {tilde over (y)} axis are the direction of antennasfixed to the rotating tool, and ψ is the tool's rotation angle, then

$\begin{matrix}{\begin{bmatrix}V_{\overset{\sim}{x}\overset{\sim}{x}} & V_{\overset{\sim}{x}\overset{\sim}{y}} & V_{\overset{\sim}{x}z} \\V_{\overset{\sim}{y}\overset{\sim}{x}} & V_{\overset{\sim}{y}\overset{\sim}{y}} & V_{\overset{\sim}{y}z} \\V_{z\overset{\sim}{x}} & V_{z\overset{\sim}{y}} & V_{zz}\end{bmatrix} = {{{R(\psi)}^{tr}\begin{bmatrix}V_{xx} & V_{xy} & V_{xz} \\V_{yx} & V_{yy} & V_{yz} \\V_{zx} & V_{zy} & V_{zz}\end{bmatrix}}{R(\psi)}}} & (20) \\{{{R(\psi)} = \begin{bmatrix}{\cos \; \psi} & {{- \sin}\; \psi} & 0 \\{\sin \; \psi} & {\cos \; \psi} & 0 \\0 & 0 & 1\end{bmatrix}}{{Then},}} & (21) \\{{V_{\overset{\sim}{x}\overset{\sim}{x}} = {{\left( {{V_{aa}\cos^{2}\vartheta} + {V_{cc}\sin^{2}\vartheta}} \right){\cos^{2}\left( {\varphi - \psi} \right)}} + {V_{aa}{\sin^{2}\left( {\varphi - \psi} \right)}}}}{V_{\overset{\sim}{y}\overset{\sim}{y}} = {{\left( {{V_{aa}\cos^{2}\vartheta} + {V_{cc}\sin^{2}\vartheta}} \right){\sin^{2}\left( {\varphi - \psi} \right)}} + {V_{aa}{\cos^{2}\left( {\varphi - \psi} \right)}}}}{V_{zz} = {{V_{aa}\sin^{2}\vartheta} + {V_{cc}\cos^{2}\vartheta}}}} & (22) \\{{V_{\overset{\sim}{x}\overset{\sim}{y}} = {V_{\overset{\sim}{y}\overset{\sim}{x}} = {{- \left( {V_{aa} + V_{cc}} \right)}\sin^{2}\vartheta \; {\cos \left( {\varphi - \psi} \right)}{\sin \left( {\varphi - \psi} \right)}}}}{V_{z\overset{\sim}{x}} = {V_{\overset{\sim}{x}z} = {{- \left( {V_{aa} + V_{cc}} \right)}\cos \; \vartheta \; \sin \; {{\vartheta cos}\left( {\varphi - \psi} \right)}}}}{V_{\overset{\sim}{y}z} = {V_{z\overset{\sim}{y}} = {{- \left( {V_{aa} - V_{cc}} \right)}\cos \; {{\vartheta sin\vartheta sin}\left( {\varphi - \psi} \right)}}}}{{The}\mspace{14mu} {following}\mspace{14mu} {relations}\mspace{14mu} {apply}\text{:}}} & (23) \\{{{V_{\overset{\sim}{x}\overset{\sim}{x}} + V_{\overset{\sim}{y}\overset{\sim}{y}} + V_{zz}} = {{2V_{aa}} + V_{cc}}}{{V_{\overset{\sim}{x}\overset{\sim}{x}} - V_{\overset{\sim}{y}\overset{\sim}{y}}} = {\left( {V_{cc} - V_{aa}} \right)\sin^{2}\vartheta \left\{ {{\cos^{2}\left( {\varphi - \psi} \right)} - {\sin^{2}\left( {\varphi - \psi} \right)}} \right\}}}{{V_{\overset{\sim}{y}\overset{\sim}{y}} - V_{zz}} = {{- \left( {V_{cc} - V_{aa}} \right)}\left\{ {{\cos^{2}\vartheta} - {\sin^{2}\vartheta \; {\sin^{2}\left( {\varphi - \psi} \right)}}} \right\}}}{{V_{zz} - V_{\overset{\sim}{x}\overset{\sim}{x}}} = {\left( {V_{cc} - V_{aa}} \right)\left\{ {{\cos^{2}\vartheta} - {\sin^{2}\vartheta \; {\cos^{2}\left( {\varphi - \psi} \right)}}} \right\}}}{{Consequently},}} & (24) \\{{{\varphi - \psi} = {\frac{1}{2}\tan^{- 1}\frac{V_{\overset{\sim}{x}\overset{\sim}{y}} + V_{\overset{\sim}{y}\overset{\sim}{x}}}{V_{\overset{\sim}{x}\overset{\sim}{x}} + V_{\overset{\sim}{y}\overset{\sim}{y}}}}}{{\varphi - \psi} = {{\tan^{- 1}\frac{V_{\overset{\sim}{y}z}}{V_{\overset{\sim}{x}z}}} = {\tan^{- 1}\frac{V_{z\overset{\sim}{y}}}{V_{z\overset{\sim}{x}}}}}}} & (25)\end{matrix}$

The azimuth angle φ is measured from the tri-axial responses if the toolrotation angle ψis known. To the contrary, the dip (deviation) angle νis determined by

$\begin{matrix}{{\tan \; \vartheta} = \sqrt{\left( \frac{V_{\overset{\sim}{x}\overset{\sim}{y}}}{V_{\overset{\sim}{x}z}} \right)^{2} + \left( \frac{V_{\overset{\sim}{x}\overset{\sim}{y}}}{V_{\overset{\sim}{y}z}} \right)^{2}}} & (26)\end{matrix}$

without knowing the tool orientation ψ.

Apparent Dip Angle and Azimuth Angle and the Distance to the Anomaly

The dip and the azimuth angle described above indicate the direction ofa resistivity anomaly determined by a combination of tri-axial transientresponses at a time (t) when the angles have deviated from a zero value.When t is small or close to zero, the effect of such anomaly is notapparent in the transient responses as all the cross-component responsesare vanishing. To identify the anomaly and estimate not only itsdirection but also the distance, it is useful to define the apparentazimuth angle φ_(app)(t) by,

$\begin{matrix}{{{\varphi_{app}(t)} = {\frac{1}{2}\tan^{- 1}\frac{{V_{xy}(t)} + {V_{yx}(t)}}{{V_{xx}(t)} - {V_{yy}(t)}}}}{{\varphi_{app}(t)} = {{\tan^{- 1}\frac{V_{yz}(t)}{V_{xz}(t)}} = {\tan^{- 1}\frac{V_{zy}(t)}{V_{zx}(t)}}}}} & (27)\end{matrix}$

and the effective dip angle ν_(app)(t) by

$\begin{matrix}{{\tan \; {\vartheta_{app}(t)}} = \sqrt{\left( \frac{V_{xy}(t)}{V_{xz}(t)} \right)^{2} + \left( \frac{V_{xy}(t)}{V_{yz}(t)} \right)^{2}}} & (28)\end{matrix}$

for the time interval when φ_(app)(t)≠0 nor π/2 (90°). For simplicity,the case examined below is one in which none of the cross-componentmeasurements is identically zero: V_(xy)(t)≠0, V_(yz)(t)≠0, andV_(zx)(t)≠0.

For the time interval when φ_(app)(t)=0, ν_(app)(t) is defined by,

$\begin{matrix}{{\vartheta_{app}(t)} = {{- \frac{1}{2}}\tan^{- 1}\frac{{V_{xz}(t)} + {V_{zx}(t)}}{{V_{xx}(t)} - {V_{zz}(t)}}}} & (29)\end{matrix}$

For the time interval when φ_(app)(t)=π/2 (90°), ν_(app)(t) is definedby,

$\begin{matrix}{{\vartheta_{app}(t)} = {{- \frac{1}{2}}\tan^{- 1}\frac{{V_{yz}(t)} + {V_{zy}(t)}}{{V_{yy}(t)} - {V_{zz}(t)}}}} & (30)\end{matrix}$

When t is small and the transient responses do not see the effect of aresistivity anomaly at distance, the effective angles are identicallyzero, φ_(app)(t)=ν_(app)(t)=0. As t increases, when the transientresponses see the effect of the anomaly, φ_(app)(t) and ν_(app)(t) beginto show the true azimuth and the true dip angles. The distance to theanomaly may be indicated at the time when φ_(app)(t) and ν_(app)(t)start deviating from the initial zero values. As shown below in amodeling example, the presence of an anomaly is detected much earlier intime in the effective angles than in the apparent conductivity(σ_(app)(t)). Even if the resistivity of the anomaly may not be knownuntil σ_(app)(t) is affected by the anomaly, its presence and thedirection can be measured by the apparent angles. With limitation intime measurement, the distant anomaly may not be seen in the change ofσ_(app)(t) but is visible in φ_(app)(t) and ν_(app)(t).

FIRST MODELING EXAMPLE

FIG. 6 depicts a simplified modeling example wherein a resistivityanomaly A is depicted in the form of, for example, a massive salt domein a formation 5. The salt interface 55 may be regarded as a planeinterface. FIG. 6 also indicates coaxial 60, coplanar 62, andcross-component (64) measurement arrangements, wherein a transmittercoil and a receiver coil are spaced a distance L apart from each other.It will be understood that in a practical application, separate toolsmay be employed for each of these arrangements, or a multiple orthogonaltool. For further simplification, it can be assumed that the azimuthdirection of the salt face as seen from the tool is known. Accordingly,the remaining unknowns are the first distance D₁ to the salt face 55from the tool, the second distance D₂ of the other side of the salt fromthe tool, the isotropic or anisotropic formation resistivity, and theapproach angle (or dip angle) θ as shown in FIG. 6. The thickness Δ ofthe salt dome is defined as Δ=D₂−D₁. In case the resistivity in theanomaly A is anisotropic, the electromagnetic properties of the anomalymay be characterized by normal resistivity R_(⊥) in the direction of theprincipal axis of the anisotropy (or normal conductivity σ_(⊥)), andin-plane resistivity R_(//) (or in-plane conductivity σ_(//)) in anydirection within a plane perpendicular to the principal axis. In case ofanisotropy, R_(//≠R) _(⊥).

Before discussing anisotropy in more detail, isotropic formations willfirst be illustrated with resistivity R(=R_(//)=R_(⊥)) (or its inverseσ=σ_(//)=σ_(⊥)).

FIG. 7 and FIG. 8 show the calculated transient voltage response (V)from coaxial V_(zz)(t) (line 65), coplanar V_(xx)(t) (line 66), andcross-component V_(zx)(t) (line 67) measurements for a tool having L=1m, for θ=30°, and located at a distance of D₁=10 m respectively D₁=100 maway from a salt face 55. In the calculations, D₂ has been assumed muchlarger than 100 m, such that within the timescale of the calculation (upto 1 sec) any influence from the other side of the salt A is notdetectable in the transient response. Moreover, when the anomaly islarge and distant compared to the transmitter-receiver spacing L, theeffect of the spacing L can be ignored and the transient responses canbe approximated with those of the receivers near the transmitter.

The effect of the resistivity anomaly A (as depicted in FIG. 6) is seenin the calculated transient responses as time increases. In addition tothe coaxial and coplanar responses (65, 66), the cross-componentresponses Vij(t) (i≠j; I, j=x, y, z) become non-zero. In order tofacilitate analysis of the responses, they may be converted to apparentdip and/or apparent conductivity.

The apparent dip angle θ_(app)(t), as calculated by

$\begin{matrix}{{{\theta_{app}(t)} = {{- \frac{1}{2}}\tan^{- 1}\frac{{V_{zx}(t)} + {V_{xz}(t)}}{{V_{zz}(t)} - {V_{xx}(t)}}}},} & (31)\end{matrix}$

is shown in FIG. 9 for a L=1 m tool assembly when the salt face 55 isD₁=10 m away and at the approach angle of θ=30°.

The apparent conductivity (σ_(app)(t)) from both the coaxial (V_(zz)(t)of FIG. 7) and the coplanar (V_(xx)(t) of FIG. 7) responses are shown inFIG. 10 (lines 68, respectively line 69), wherein the approach angle(θ=30°) and salt face distance (D₁=10 m) are the same as in FIG. 9.Details of how the apparent conductivities are calculated will beprovided below.

Note that the true direction from the tool to the salt face (i.c. 30°)is reflected in the apparent dip θ_(app)(t) plot of FIG. 9 as early as10⁻⁴ second, when the presence of the resistivity anomaly is barelydetected in the apparent conductivity (σ_(app)(t)) plot of FIG. 10. Ittakes almost 10⁻³ second for the apparent conductivity to approach anasymptotic σ_(app)(late t) value.

FIG. 11 shows the apparent dip θ_(app)(t) for the L=1 m tool assemblywhen the salt face is D=10 m away, but at different angles between thetool axis and the target varying from 0 to 90° in 15° increments. Theapproach angle (θ) may be reflected at any angle in about 10⁻⁴ sec.

FIG. 11 and FIGS. 12 and 13 compare the apparent dip θ_(app)(t) fordifferent salt face distances (D=10 m; 50 m; and 100 m) and differentangles between the tool axis and the target.

The distance to the salt face can be also determined by the transitiontime at which θ_(app)(t) takes an asymptotic value. Even if the saltface distance (D) is 100 m, it can be identified and its direction canbe measured by the apparent dip θ_(app)(t).

In summary, the method considers the coordinate transformation oftransient EM responses between tool-fixed coordinates and anomaly-fixedcoordinates. When the anomaly is large and far away compared to thetransmitter-receiver spacing, one may ignore the effect of spacing andapproximate the transient EM responses with those of the receivers nearthe transmitter. Then, one may assume axial symmetry exists with respectto the c-axis that defines the direction from the transmitter to theanomaly. In such an axially symmetric configuration, the cross-componentresponses in the anomaly-fixed coordinates are identically zero. Withthis assumption, a general method is provided for determining thedirection to the resistivity anomaly using tri-axial transient EMresponses.

The method defines the apparent dip θ_(app)(t) and the apparent azimuthφ_(app)(t) by combinations of tri-axial transient measurements. Theapparent direction {θ_(app)(t), φ_(app)(t)} reads the true direction {θ,φ} at later time. The θ_(app)(t) and φ_(app)(t) both read zero when t issmall and the effect of the anomaly is not sensed in the transientresponses or the apparent conductivity. The conductivities(σ_(coaxial)(t) and σ_(coplanar)(t)) from the coaxial and coplanarmeasurements both indicate the conductivity of the near formation aroundthe tool.

Deviation of the apparent direction ({θ_(app)(t), φ_(app)(t)}) from zeroidentifies the anomaly. The distance to the anomaly is measured by thetime when the apparent direction ({θ_(app)(t), φ_(app)(t)}) starts todeviate from zero or by the time when the apparent direction({θ_(app)(t), φ_(app)(t)}) starts approaches the true direction ({θ,φ}). The distance can be also measured from the change in the apparentconductivity. However, the anomaly is identified and measured muchearlier in time in the apparent direction than in the apparentconductivity.

Apparent Conductivity

As set forth above, apparent conductivity can be used as an alternativetechnique to apparent angles in order to determine the location of ananomaly in a wellbore. The time-dependent apparent conductivity can bedefined at each point of a time series at each logging depth. Theapparent conductivity at a logging depth z is defined as theconductivity of a homogeneous formation that would generate the sametool response measured at the selected position.

In transient EM logging, transient data are collected at a logging depthor tool location z as a time series of induced voltages in a receiverloop. Accordingly, time dependent apparent conductivity (σ(z; t)) may bedefined at each point of the time series at each logging depth, for aproper range of time intervals depending on the formation conductivityand the tool specifications.

The induced voltage of a coaxial tool with transmitter-receiver spacingL in the homogeneous formation of conductivity (σ) is given by,

$\begin{matrix}{{V_{zZ}(t)} = {{C\frac{\left( {\mu_{o}\sigma} \right)^{3/2}}{8t^{5/2}}^{- u^{2}}\mspace{14mu} {where}\mspace{14mu} u^{2}} = {\frac{\mu_{o}\sigma}{4}\frac{L^{2}}{t}}}} & (32)\end{matrix}$

and C is a constant.

The time-changing apparent conductivity depends on the voltage responsein a coaxial tool (V_(zZ)(t)) at each time of measurement as:

$\begin{matrix}{{C\frac{\left( {\mu_{o}{\sigma_{app}(t)}} \right)^{3/2}}{8t^{5/2}}^{- {u_{app}{(t)}}^{2}}} = {{{V_{zZ}(t)}\mspace{14mu} {where}\mspace{14mu} {u_{app}(t)}^{2}} = {\frac{\mu_{o}{\sigma_{app}(t)}}{4}\frac{L^{2}}{t}}}} & (33)\end{matrix}$

and V_(zZ)(t) on the right hand side is the measured voltage response ofthe coaxial tool. From a single type of measurement (coaxial, singlespacing), the greater the spacing L, the larger the measurement time (t)should be to apply the apparent conductivity concept. The σ_(app)(t)should be constant and equal to the formation conductivity in ahomogeneous formation: σ_(app)(t)=σ. The deviation from a constant (σ)at time (t) suggests a conductivity anomaly in the region specified bytime (t).

The induced voltage of the coplanar tool with transmitter-receiverspacing L in the homogeneous formation of conductivity (σ) is given by,

$\begin{matrix}{{V_{xX}(t)} = {{C\frac{\left( {\mu_{0}\sigma} \right)^{3/2}}{8t^{5/2}}\left( {1 - u^{2}} \right)^{- u^{2}}\mspace{14mu} {where}\mspace{14mu} u^{2}} = {\frac{\mu_{o}\sigma}{4t}L^{2}}}} & (34)\end{matrix}$

and C is a constant. At small values of t, the coplanar voltage changespolarity depending on the spacing L and the formation conductivity.

Similarly to the coaxial tool response, the time-changing apparentconductivity is defined from the coplanar tool response V_(xX)(t) ateach time of measurement as,

$\begin{matrix}{{C\frac{\left( {\mu_{o}{\sigma_{app}(t)}} \right)^{3/2}}{8t^{5/2}}\left( {1 - {u_{app}(t)}^{2}} \right)^{- {u_{app}{(t)}}^{2}}} = {{{V_{xX}(t)}\mspace{14mu} {where}\mspace{14mu} {u_{app}(t)}^{2}} = {\frac{\mu_{o}{\sigma_{app}(t)}}{4}\frac{L^{2}}{t}}}} & (35)\end{matrix}$

and V_(xX)(t) on the right hand side is the measured voltage response ofthe coplanar tool. The longer the spacing, the larger the value t shouldbe to apply the apparent conductivity concept from a single type ofmeasurement (coplanar, single spacing). The σ_(app)(t) should beconstant and equal to the formation conductivity in a homogeneousformation: σ_(app)(t)=σ.

When there are two coaxial receivers, the ratio between the pair ofvoltage measurements is given by,

$\begin{matrix}{\frac{V_{zZ}\left( {L_{1};t} \right)}{V_{zZ}\left( {L_{2};t} \right)} = ^{{- \frac{\mu_{o}\sigma}{4t}}{({L_{1}^{2} - L_{2}^{2}})}}} & (36)\end{matrix}$

where L₁ and L₂ are transmitter-receiver spacing of two coaxial tools.

Conversely, the time-changing apparent conductivity is defined for apair of coaxial tools by,

$\begin{matrix}{{\sigma_{app}(t)} = {\frac{- {\ln \left( \frac{V_{zZ}\left( {L_{1};t} \right)}{V_{zZ}\left( {L_{2};t} \right)} \right)}}{\left( {L_{1}^{2} - L_{2}^{2}} \right)}\frac{4t}{\mu_{o}}}} & (37)\end{matrix}$

at each time of measurement. The σ_(app)(t) should be constant and equalto the formation conductivity in a homogeneous formation: σ_(app)(t)=σ.

The apparent conductivity is similarly defined for a pair of coplanartools or for a pair of coaxial and coplanar tools. The σ_(app)(t) shouldbe constant and equal to the formation conductivity in a homogeneousformation: σ_(app)(t)=σ. The deviation from a constant (σ) at time (t)suggests a conductivity anomaly in the region specified by time (t).

As will be illustrated below, apparent conductivity (σ_(app)(t)),whether coaxial or coplanar, may reveal three parameters in relation toa two-layer formation, including:

(1) the conductivity of a local first layer in which the tool islocated;(2) the conductivity of one or more adjacent layers or beds; and(3) the distance of the tool to the layer boundaries.

Analysis of Coaxial Transient Response in Two-Layer Models

To illustrate usefulness of the concept of apparent conductivity, thetransient response of a tool in a two-layer earth model, as in FIG. 14for example, can be examined.

FIG. 14 illustrates a coaxial tool 80 in which both a transmitter coil(T) and a receiver coil (R) are wound around the common tool axis z andspaced a distance L apart. The symbols σ₁ and σ₂ may represent theconductivities of two formation layers. The coaxial tool 80 be placed ina horizontal well 88 traversing formation layer 5 and extending parallelto the layer interface 55.

In the present example, a horizontal well is depicted such that thedistance from the tool to the layer boundary corresponds to the distanceof the horizontal borehole to the layer boundary. Under a more generalcircumstance, the relative direction of a borehole and tool to the bedinterface is not known.

The calculated transient voltage response V(t) for the L=1 mtransmitter-receiver offset coaxial tool at various distances D betweenthe tool 80 and the layer boundary 55 is shown in FIG. 15 for D=1, 5,10, 25, and 50 m. The formation can be analyzed using these responses,employing apparent conductivity as further explained with regard toFIGS. 16 and 17.

FIG. 16 shows the voltage data of FIG. 15 plotted in terms of apparentconductivity, for a geometry wherein σ₁=0.1 S/m (R₁=10 Ωm) and σ₂=1 S/m(R₂=1 Ωm). Similarly, FIG. 17 illustrates the apparent conductivity in atwo-layer model where σ₁=1 S/m (R₁=1 Ωm) and σ₂=0.1 S/m (R₂=10 Ωm).

The apparent conductivity plots reveal a “constant” conductivity atsmall t, and at large t but having a different value, and a transitiontime t_(c) that marks the transition between the two “constant”conductivity values and depends on the distance D.

As will be further explained below, in a two-layer resistivity profile,the apparent conductivity as t approaches zero can identify the layerconductivity σ₁ around the tool, while the apparent conductivity as tapproaches infinity can be used to determine the conductivity σ₂ of theadjacent layer at a distance. The distance to the bed boundary 55 fromthe tool 80 can also be measured from the transition time t_(c) observedin the apparent conductivity plots.

At small values of t, the tool reads the apparent conductivity σ₁ of thefirst layer 5 around the tool 80. Conductivity at small values of t isthought to correspond to the conductivity of the local layer 5 where thetool is located in. At small values of t, the signal reaches thereceiver directly from the transmitter without interfering with the bedboundary. Namely, the signal is affected only by the conductivity σ₁around the tool.

At large values of t, the tool reads 0.4 S/m for a two-layer model whereeither σ₁=1 S/m (R₁=1 Ωm) and σ₂=0.1 S/m (R₂=10 Ωm), or σ₁=0.1 S/m(R₁=10 Ωm) and σ₂=1 S/m (R₂=1 Ωm). The value of 0.4 is believed tocorrespond to some average between the conductivities of the two layers,because at large values of t, nearly half of the signals come from theformation below the tool and the remaining signals come from above, ifthe time for the signal to travel the distance between the tool and thebed boundary is small.

This is further investigated in FIG. 18, which shows examples of theσ_(app)(t) plots for D=1 m and L=1 m, but for different resistivityratios of the target layer 2 while the local conductivity (σ₁) is fixedat 1 S/m (R₁=1 Ωm). The apparent conductivity at large values of t isdetermined by the target layer 2 conductivity, as shown in line 71 inFIG. 19 when σ₁ is fixed at 1 S/m.

Numerically, the late time conductivity may be approximated by thesquare root average of two-layer conductivities as:

$\begin{matrix}{\sqrt{\sigma_{app}\left( {{{t->\infty};\sigma_{1}},\sigma_{2}} \right)} = {\frac{\sqrt{\sigma_{1}} + \sqrt{\sigma_{2}}}{2}.}} & (39)\end{matrix}$

This is depicted as line 72 in FIG. 19.

Thus, the conductivity at large values of t (as t approaches infinity)can be used to estimate the conductivity (σ₂) of the adjacent layer whenthe local conductivity (σ₁) near the tool is known, for instance fromthe conductivity as t approaches zero as illustrated in FIG. 20.

Estimation of D, The Distance to the Electromagnetic Anomaly

The distance D from the tool to the bed is reflected in the transitiontime t_(c). The transition time at which the apparent conductivity(σ_(app)(t)) starts deviating from the local conductivity (σ₁) towardsthe conductivity at large values of t depends on D and L, as shown inFIG. 21.

For convenience, the transition time (t_(c)) can be defined as the timeat which the σ_(app)(t_(c)) takes a cutoff conductivity (σ_(c)). In thiscase, the cutoff conductivity is represented by the arithmetic averagebetween the conductivity as t approaches zero and the conductivity as tapproaches infinity. The transition time (t_(c)) is dictated by the raypath RP:

$\begin{matrix}{{{RP} = \sqrt{\left( \frac{L}{2} \right)^{2} + D^{2}}},} & (40)\end{matrix}$

that is the shortest distance for the electromagnetic signal travelingfrom the transmitter to the bed boundary, to the receiver, independentlyof the resistivity of the two layers. Conversely, the distance (D) tothe anomaly can be estimated from the transition time (t_(c)), as shownin FIG. 22.

Look-Ahead Capabilities of EM Transient Method

By analyzing apparent conductivity or its inherent inverse equivalent(apparent resistivity), the present invention can identify the locationof a resistivity anomaly (e.g., a conductive anomaly and a resistiveanomaly). Further, resistivity or conductivity can be determined fromthe coaxial and/or coplanar transient responses. As explained above, thedirection to the anomaly can be determined if the cross-component dataare also available. To further illustrate the usefulness of theseconcepts, the foregoing analysis may also be used to detect an anomalyat a distance ahead of the drill bit.

FIG. 23 shows a coaxial tool 80 with transmitter-receiver spacing Lplaced in, for example, a vertical well 88 approaching or just beyond anadjacent bed that is a resistivity anomaly. The tool 80 includes both atransmitter coil T and a receiver coil R, which are wound around acommon tool axis and are oriented in the tool axis direction. Thesymbols σ₁ and σ₂ may represent the conductivities of two formationlayers, and D the distance between the tool 80 (e.g. the transitionantenna T) and the layer boundary 55.

The calculated transient voltage response of the L=1 m(transmitter-receiver offset) coaxial tool at different distances (D=1,5, 10, 25, and 50 m) as a function of t is shown in FIG. 24, in a casewherein σ₁=0.1 S/m (corresponding to R₁=10 Ωm), and σ₂=1 S/m(corresponding to R₂=1 Ωm). Though difference is observed amongresponses at different distances, it is not straightforward to identifythe resistivity anomaly directly from these responses.

The same voltage data of FIG. 24 is plotted in terms of the apparentconductivity (σ_(app)(t)) in FIG. 25. From this Figure, it is clear thatthe coaxial response can identify an adjacent bed of higher conductivityat a distance. Even a L=1 m tool can detect the bed at 10, 25, and 50 maway, if low voltage response can be measured for 0.1-1 seconds long.

The σ_(app)(t) plot exhibits at least three parameters very distinctlyin the figure: the early time conductivity; the later time conductivity;and the transition time that moves as the distance (D) changes. In FIG.25, it should be noted that, at early time whereby t is close to zero,the tool reads the apparent conductivity of 0.1 S/m, which isrepresentative of the layer just around the tool. The signal thatreaches the receiver R not yet contains information about the boundary55. At later time, the tool reads close to 0.55 S/m, representing anarithmetic average between the conductivities of the two layers. Atlater time, t→∞, nearly half of the signals come from the formationbelow the tool and the other half from above the tool, if the time totravel the distance (D) of the tool to the bed boundary is small. Thedistance D is reflected in the transition time t_(c).

FIG. 26 illustrates the σ_(app)(t) plot of the coaxial transientresponse in the two-layer model of FIG. 23 for an L=1 m tool atdifferent distances (D), except that the conductivity of the local layer(σ₁) is 1 S/m (R₁=1 Ωm) and the conductivity of the target layer (σ₂) is0.1 S/m (R₂=10 Ωm). Again, the tool reads at early time the apparentconductivity of 1.0 S/m that is of the layer just around the tool. At alater time, the tool reads about 0.55 S/m, the same average conductivityvalue as in FIG. 25. The distance (D) is reflected in the transitiontime t_(c).

Hence, the transient electromagnetic response method can be used as alook-ahead resistivity logging method.

FIG. 27 compares the σ_(app)(t) plot of FIG. 25 and FIG. 26 for L=1 mand D=50 m. The late time conductivity is determined solely by theconductivities of the two layers (σ₁ and σ₂) alone. It is not affectedby where the tool is located in the two layers. However, because of thedeep depth of investigation, the late time conductivity is not readilyreached even at t=1 second, as shown in FIG. 25. In practice, the latetime conductivity may have to be approximated by σ_(app)(t=1 second)which slightly depends on D as illustrated in FIG. 25.

Numerically, the late time apparent conductivity may be approximated bythe arithmetic average of two-layer conductivities as:

${\sigma_{app}\left( {{{t->\infty};\sigma_{1}},\sigma_{2}} \right)} = {\frac{\sigma_{1} + \sigma_{2}}{2}.}$

This is reasonable considering that, with the coaxial tool, the axialtransmitter induces the eddy current parallel to the bed boundary. Atlater time, the axial receiver receives horizontal current nearlyequally from both layers. As a result, the late time conductivity mustsee conductivity of both formations with nearly equal weight.

FIG. 28 compares the σ_(app)(t) plots for D=50 m but with differentspacing L. The σ_(app)(t) reaches a nearly constant late time apparentconductivity at later times as L increases. The late time apparentconductivity (σ_(app)(t→∞) is nearly independent of L. However, the latetime conductivity defined at t=1 second, depends on slightly thedistance (D).

Thus, the late time apparent conductivity (σ_(app)(t→∞)) at t=1 secondcan be used to estimate the conductivity of the adjacent layer (σ₂) whenthe local conductivity near the tool (σ₁) is known, for instance, fromthe early time apparent conductivity (σ_(app)(t→0)=σ₁).

Estimation of the Distance (D) to the Electromagnetic Anomaly

The transition time (t_(c)) at which the apparent conductivity startsdeviating from the local conductivity (σ₁) toward the late timeconductivity clearly depends on D, the distance of the tool to the bedboundary, as shown in FIG. 25 for a L=1 m tool.

For convenience, the transition time (t_(c)) is defined by the time atwhich the σ_(app)(t_(c)) takes the cutoff conductivity (σ_(c)), that is,in this example, the arithmetic average between the early time and thelate time conductivities: σ_(c)={σ_(app)(t→0)+σ_(app)(t→∞)}/2. Thetransition time (t_(c)) is dictated by the ray-path RP, D minus L/2 thatis, half the distance for the EM signal to travel from the transmitterto the bed boundary to the receiver, independently on the resistivity ofthe two layers. Conversely, the distance (D) can be estimated from thetransition time (t_(c)), as shown in FIG. 29 when L=1 m.

Analysis of Coplanar Transient Responses in Two-Layer Models

While the coaxial transient data were examined above, the coplanartransient data are equally useful as a look-ahead resistivity loggingmethod.

FIG. 30 shows a coplanar tool 80 with transmitter-receiver spacing Lplaced in a well 88 and approaching (or just beyond) layer boundary 55of an adjacent bed that is the resistivity anomaly. On the coplanartool, both a transmitter T and a receiver R are oriented perpendicularlyto the tool axis z and parallel to each other. The symbols σ₁, and σ₂may represent the conductivities of two formation layers.

Corresponding to FIG. 25 for coaxial tool responses where L=1 m, theapparent conductivity (σ_(app)(t)) for calculated coplanar responses isplotted in FIG. 31 for different tool distances from the bed boundary55. It is clear that the coplanar response can also identify an adjacentbed of higher conductivity at a distance. Even a L=1 m tool can detectthe bed at 10 m, 25 m, and 50 m away if low voltage responses can bemeasured for 0.1-1 seconds long. The σ_(app)(t) plot for the coplanarresponses exhibits three parameters equally as well as for the coaxialresponses.

Like it was the case in the co-axial geometry, it is also true for thecoplanar responses that the early time apparent conductivity(σ_(app)(t→0)) is the conductivity of the local layer (σ₁) where thetool is located. Conversely, the layer conductivity can be measuredeasily by the apparent conductivity at earlier times.

The late time apparent conductivity (σ_(app)(t→∞)) is some average ofconductivities of both layers. The conclusions derived for the coaxialresponses apply equally well to the coplanar responses. However, thevalue of the late time conductivity for the coplanar responses is notthe same as for the coaxial responses. For coaxial responses, the latetime conductivity is close to the arithmetic average of two-layerconductivities in two-layer models.

FIG. 32 shows the late time conductivity (σ_(app)(t→∞)) for coplanarresponses as obtained from the model calculations (line 77) whereby D=50m and L=1 m, but for different conductivities of the local layer whilethe target conductivity is fixed at 1 S/m. Late time conductivity isdetermined by the local layer conductivity, and is numerically close tothe square root average as,

${\sqrt{\sigma_{app}\left( {{{t->\infty};\sigma_{1}},\sigma_{2}} \right)} = \frac{\sqrt{\sigma_{1}} + \sqrt{\sigma_{2}}}{2}},$

as is shown by line 78 in FIG. 32.

To summarize, the late time conductivity (σ_(app)(t→∞)) can be used toestimate the conductivity of the adjacent layer (σ₂) when the localconductivity near the tool (σ₁) is known, for instance, from the earlytime conductivity (σ_(app)(t→0)=σ₁). This is illustrated in FIG. 33wherein line 79 has been obtained from model calculations and line 79 adisplays the average approximation.

Estimation of the Distance (D) to the Electromagnetic Anomaly

The transition time t_(c) at which the apparent conductivity startsdeviating from the local conductivity (σ₁) toward the late timeconductivity clearly depends on the distance (D) of the tool 80 (e.g.the transmitter T) to the bed boundary 55, as shown in FIG. 30.

The transition time (t_(c)) may be defined by the time at which theσ_(app)(t_(c)) takes the cutoff conductivity (σ_(c)) that is, in thisexample, the arithmetic average between the early time and the late timeconductivities: σ_(c)={σ_(app)(t→0)+σ_(app)(t→∞)}/2. The transition time(t_(c)) is dictated by the ray-path, D minus L/2 that is, half thedistance for the EM signal to travel from the transmitter to the bedboundary to the receiver, independently of the resistivity of the twolayers.

Conversely, the distance (D) can be estimated from the transition time(t_(c)), as shown in FIG. 34, where L=1 m.

Analysis of Transient Electromagnetic Response Data for Three or MoreFormation Layers

The next model shows a conductive near layer, a very resistive layer,and a further conductive layer. The geological configuration is depictedin FIG. 35, together with a coaxial tool 80 in a relatively conductiveformation 82 wherein an anomaly is located in the form of a relativelyresistive layer 83. As shown, the formation on the other side of layer83, as seen from tool 80 and identified in FIG. 35 by reference numeral84, is identical to the formation 82 on the tool side of the layer 83.However, the method will also work if the formation 84 on the other sideof layer 83 would constitute a layer that has different properties fromthose of the near formation 82.

In either case, the tool “sees” the anomaly 83 as a first layer at afirst distance D1 away and having a thickness Δ, and it “sees” theformation on the other side of the anomaly 83 as a second layer 84 at asecond distance D₂=D₁+Δ away and having infinite thickness. FIG. 36 is agraph showing calculated apparent resistivity response R_(app) versustime t for a geometry as given in FIG. 35. For the calculation of FIG.36, it has been assumed that the anomaly is formed of a resistive saltbed, having a resistivity of 100 Ωm, and that the formation is formed offor instance a brine-saturated formation having a resistivity of 1 Ωm.The tool has been modeled as being oriented with its main axis parallelto the first interface 81 between the brine-saturated formation 82, andthe distance between the main axis and the first layer 83, D₁, has beentaken 10 m. The resistive bed thickness Δ has been varied from afraction of a to 100 meters in thickness.

The first climb of R_(app)(t) is the response to the salt and takesplace at 10⁻⁴ s with an L=1 m tool when the salt is at D₁=10 m away. Ifthe salt is fully resolved (by infinitely thick salt beyond D₁=10 m),the apparent resistivity should read 3 Ωm asymptotically. The subsequentdecline of R_(app)(t) is the response to a conductive formation behindthe salt (resistive bed). R_(app)(late t) is a function of conductivebed resistivity and salt thickness. If the time measurement is limitedto 10⁻² s, the decline of R_(app)(t) may not be detected for the saltthicker than 500 m.

With respect to the resistive bed resolution, the coaxial responds to athin (1-2 m thick) bed. The time at which R_(app)(t) peaks or beginsdeclining depends on the distance to the conductive bed behind the salt.As noted previously, when plotted in terms of apparent conductivityσ_(app)(t), the transition time may be used to determine the distance tothe boundary beds.

Another three-layer formation was also modeled, as shown in FIG. 37. Inthis instance, the intermediate layer 83 was a more conductive layerthan the surrounding formation 82. This conductive bed 83 may beconsidered representative of, for instance, a shale layer. The coaxialtool 80, having an L=1 m spacing, is located in a borehole in aformation 82 having a resistivity of 10 Ωm and is located D₁=10 m fromthe less resistive (more conductive) layer 83, which has a resistivityof 1 Ωm. The third layer 84 is beyond the conductive bed 83 and has aresistivity of 10 Ωm as does layer 82. The conductive bed 83 was modeledfor a range of thicknesses A varying from fractions of a meter up to aninfinite thickness. The apparent resistivity, as calculated, is setforth in FIG. 38.

The decrease in R_(app)(t), which can be seen in FIG. 38, is attributedto the presence of the shale (conductive) layer and appears as t→10⁻⁵ s.The shale response is fully resolved by an infinitely thick conductivelayer that approaches 3 Ωm. The subsequent rise in R_(app)(t) is inresponse to the resistive formation 84 beyond the shale layer 83. Thetransition time is utilized to determine the distance D₂ from the tool80 to the interface 85 between the second and third layers (83respectively 84). R_(app)(late t) is a function of conductive bedresistivity. As the conductive bed thickness Δ increases, the timemeasurement must likewise be increased (>10 ⁻² s) in order to measurethe rise of R_(app)(t) for conductive layers thicker than 100 m.

Still another three-layer model is set forth in FIG. 39, wherein thecoaxial tool 80 is in a conductive formation 82 (1 Ωm), and a highlyresistive second layer 84 (100 Ωm) as might be found in, for instance, asalt dome. Formation 82 and the second layer 84 are separated by a firstlayer 83 that has an intermediate resistance (10 Ωm). The thickness Ahas been varied in the calculations of the apparent resistivityresponse, as depicted in FIG. 40.

The response to the intermediate resistive layer is seen at 10⁻⁴ s,where R_(app)(t) increases. If the first layer 83 is fully resolved byan infinitely thick bed, the apparent resistivity approaches a 2.6 Ωmasymptote. As noted in FIG. 40, the R_(app)(t) undergoes a second stageincrease in response to the 100 Ωm highly resistive second layer 84.Based on the transition time, the distance to the interface isdetermined to be 110 m.

Though complex, the apparent resistivity or apparent conductivity in theabove examples delineates the presence of multiple layers. The observedchanges of apparent conductivity (or apparent resistivity) allowdetermination of the distances D₁ and D₂.

Transient Electromagnetic Responses Involving Formation Anisotropy

As stated above, an electromagnetic anomaly may display anisotropicelectromagnetic properties. An example is shown in FIG. 6, ifR_(//)≠R_(⊥).

Various mechanisms may give rise to a macroscopic electromagneticinduction effect. For instance, oriented fractions may generate ananisotropic response. Electromagnetic anisotropy may also ariseintrinsically in certain types of formations, such as shales, of mayarise as a result of sequences of relatively thin layers.

In the way as depicted in FIG. 6, the principal anisotropy directioncorresponds to the approach angle θ. This correspondence is mainly forreasons of simplicity in setting forth the embodiments, and need notnecessarily be the case in every situation within the scope of theinvention.

In the following it will be explained how electromagnetic anisotropy ofat least one of the formation layers may be taken into account whenanalyzing time-dependent transient response signals. This may comprisedetermining one or more anisotropy parameters that characterize theanisotropic electromagnetic properties. Amongst anisotropy parametersare anisotropy ratio α², anisotropic factor β, conductivity along aprincipal anisotropy axis σ_(⊥) (or resistivity along the principalanisotropy axis R_(⊥)), conductivity in a plane perpendicular to theprincipal anisotropy axis σ_(//) (or resistivity in a planeperpendicular to the principal anisotropy axis R_(//)); tool axis anglerelative to the principal anisotropy axis.

Using the concepts of apparent conductivity or apparent resistivityand/or apparent dip or azimuth, the distance and/or direction to ananomaly may be determined from the time-dependent transient responsesignals even when the anomaly, and/or a distant formation layer,comprise(s) an electromagnetic anisotropy or when the transmitter and/orreceiver antennae are embedded in an anisotropic formation layer.

Using the principles set forth above, the analysis taking into accountanisotropy may be extended to multiple bedded formations, includingthose where only a distant formation layer or target anomaly givesanisotropic electromagnetic induction responses (such as for instance inFIG. 6) or where a local formation layer wherein the transmitter andreceiver antennae are located, displays anisotropic behavior and one ormore other, isotropic or anisotropic layers are present at a distance.The distance and direction from the tool to the more distant layersand/or the target anomaly may then be determined, provided thatanisotropy is taken into account.

In the forthcoming explanation, for reasons of simplicity, it will beassumed that the anisotropy has a vertically aligned principal axis,such that the angle between the tool axis z and the principal anisotropyaxis corresponds to the dip angle or deviation angle θ. The termhorizontal resistivity R_(H) may be employed, which generallycorresponds to the resistivity in the anisotropy plane perpendicular tothe principal anisotropy direction. The term vertical resistivity R_(V)generally refers to resistivity in the principal anisotropy direction ornormal direction.

Transient EM Responses in a Homogeneous Anisotropic Formation

Considered is an anisotropic formation, in which a vertical resistivityR_(V) (or its inverse vertical conductivity σ_(V)) is different from thehorizontal resistivity R_(H) (or horizontal conductivity σ_(H)). Assumedis that the formation is azimuth-symmetric, in the horizontal direction.The tool axis z is deviated from the vertical direction by the dip(deviation) angle θ in the zx-plane. The transmitter antenna is placedat origin. The receiver antenna is placed at (x=L·sin θ, y=0, z=L·cosθ). There may be four independent combinations of transmitter andreceiver orientations that render non-zero responses.

In addition to a coaxial response, V_(Zz), there are two coplanarresponses, V_(Xx) and V_(Yy), and one cross-component responseV_(Xz)=V_(Zx). One coplanar response, V_(Xx), is from a transversetransmitter antenna and receiver antenna that are oriented within thezx-plane. Another coplanar response, V_(Yy), is from a transversetransmitter and receiver both of which are oriented in the y-axisdirection. The cross-component response is from a transverse receiverantenna with the longitudinally oriented transmitter antenna, or viseversa. The transverse receiver antenna is directed within the zx-plane.Any cross-component involving either a transmitter or a receiveroriented in the y-axis direction, i.e. V_(Yx) and V_(xy) and V_(Yz) andV_(Zy) are all vanishing.

The above has been set forth in tool-coordinates. It is further remarkedthat any antenna that is sensitive to a transverse component of anelectromagnetic induction field suffices as a transverse antenna.

Applicants have derived the transient response in time domain, expressedin terms of horizontal conductivity σ_(H) and anisotropic factor β, aregiven by:

$\begin{matrix}{{{V_{Zz}(t)} = {C\frac{\left( {\mu_{0}\sigma_{H}} \right)^{3/2}}{8t^{5/2}}^{- u^{2}}\begin{Bmatrix}{1 + {\frac{1}{2}\left( {{\beta^{2}^{- {u^{2}{({\beta^{2} - 1})}}}} - 1} \right)} -} \\{\frac{1}{4u^{2}}\left( {^{- {u^{2}{({\beta^{2} - 1})}}} - 1} \right)}\end{Bmatrix}}};} & (41) \\{{{V_{Xx}(t)} = {C\frac{\left( {\mu_{0}\sigma_{H}} \right)^{3/2}}{8t^{5/2}}^{- u^{2}}\begin{Bmatrix}{\left\lbrack {1 - u^{2}} \right\rbrack + \frac{\cos^{2}\theta}{\sin^{2}\theta}} \\\begin{bmatrix}{{\frac{1}{2}\left( {{\beta^{2}^{- {u^{2}{({\beta^{2} - 1})}}}} - 1} \right)} -} \\{\frac{1}{4u^{2}}\left( {^{- {u^{2}{({\beta^{2} - 1})}}} - 1} \right)}\end{bmatrix}\end{Bmatrix}}};} & (42) \\{{{{V_{Zx}(t)} = {{V_{Xz}(t)} = {C\frac{\left( {\mu_{0}\sigma_{H}} \right)^{3/2}}{8t^{5/2}}^{- u^{2}}\left\{ {\frac{\cos \; \theta}{\sin \; \theta}\begin{bmatrix}{{\frac{1}{2}\left( {{\beta^{2}^{- {u^{2}{({\beta^{2} - 1})}}}} - 1} \right)} -} \\{\frac{1}{4u^{2}}\left( {^{- {u^{2}{({\beta^{2} - 1})}}} - 1} \right)}\end{bmatrix}} \right\}}}};}{and}} & (43) \\{{V_{Yy}(t)} = {C\frac{\left( {\mu_{0}\sigma_{H}} \right)^{3/2}}{8t^{5/2}}^{- u^{2}}{\begin{Bmatrix}{\left\lbrack {1 - u^{2}} \right\rbrack - {\frac{1}{\sin^{2}\theta}\begin{bmatrix}{{\frac{1}{2}\left( {{\beta^{2}^{- {u^{2}{({\beta^{2} - 1})}}}} - 1} \right)} -} \\{\frac{1}{4u^{2}}\left( {^{- {u^{2}{({\beta^{2} - 1})}}} - 1} \right)}\end{bmatrix}} +} \\{\frac{1}{2}\left\lbrack {{{\alpha^{2}\left( {3 - {2u^{2}\beta^{2}}} \right)}^{- {u^{2}{({\beta^{2} - 1})}}}} - \left( {3 - {2u^{2}}} \right)} \right\rbrack}\end{Bmatrix}.}}} & (44)\end{matrix}$

In these equations,

$u^{2} = {\frac{\mu_{o}\sigma_{H}}{4}\frac{L^{2}}{t}}$

and C is a constant. The anisotropic factor β is defined as:

$\begin{matrix}{{{\beta = \sqrt{1 + {\left( {\alpha^{2} - 1} \right)\sin^{2}\theta}}};}{\alpha^{2} = {\frac{\sigma_{V}}{\sigma_{H}}.}}} & (45)\end{matrix}$

The following remarks may be made based on these equations:

1. The coaxial response depends only on the horizontal resistivityR_(H)(=1/σ_(H)) and the anisotropic factor β that is determined by theanisotropy ratio α²=σ_(V)/σ_(H)=R_(H)/R_(V), and the dip angle θ.Conversely, neither the anisotropy nor the dip angle can be determinedfrom coaxial measurements alone.

2. Both coplanar responses depend on the horizontal resistivity, theanisotropic factor, and the dip angle. 3. In vertical boreholes withθ=0, the coaxial response depends only on the horizontal resistivity,while the coplanar response is determined by both the horizontalresistivity and the vertical resistivity. 4. In horizontal logging withθ=π/2, the coaxial response depends on both the horizontal resistivityand the vertical resistivity, but the coplanar response is determinedsolely by the horizontal resistivity.

5. Because u²→0 as t→large, the dip angle is determined by:

$\begin{matrix}{{\frac{2{V_{Xz}(t)}}{{V_{Xx}(t)} - {V_{Zz}(t)}} = {{\tan \; 2\theta} + {O\left( u^{2} \right)}}},} & (46)\end{matrix}$

whereby O(u²) denotes a remainder on the order of u².

Late Time Responses in a Homogeneous Anisotropic Formation

Similar to the investigation set forth above with regard to layermodels, the late time limits may be derived. As t→∞, u²→0, and thereforethese limits converge. Taking into account anisotropy, the late timelimits of equations (41) to (44) are:

$\begin{matrix}{{{V_{Zz}(t)} = {C\frac{\left( {\mu_{0}\sigma_{H}} \right)^{3/2}}{8t^{5/2}}\left\{ {1 + {\frac{3}{4}\left( {\alpha^{2} - 1} \right)\sin^{2}\theta}} \right\}}};} & (47) \\{{{V_{Xx}(t)} = {C\frac{\left( {\mu_{0}\sigma_{H}} \right)^{3/2}}{8t^{5/2}}\left\{ {1 + {\frac{3}{4}\left( {\alpha^{2} - 1} \right)\cos^{2}\theta}} \right\}}};} & (48) \\{{{{V_{Xz}(t)} = {C\frac{\left( {\mu_{0}\sigma_{H}} \right)^{3/2}}{8t^{5/2}}\left\{ {\frac{3}{4}\left( {\alpha^{2} - 1} \right)\cos \; {\theta sin}\; \theta} \right\}}};}{and}} & (49) \\{{V_{Yy}(t)} = {C\frac{\left( {\mu_{0}\sigma_{H}} \right)^{3/2}}{8t^{5/2}}{\left\{ {1 + {\frac{3}{4}\left( {\alpha^{2} - 1} \right)}} \right\}.}}} & (50)\end{matrix}$

The dip (deviation) angle is determined by:

$\begin{matrix}{\theta = {\frac{1}{2}\tan^{- 1}{\frac{2{V_{Xz}(t)}}{{V_{Xx}(t)} - {V_{Zz}(t)}}.}}} & (51)\end{matrix}$

The anisotropy ratio α² may be determined from:

$\begin{matrix}{\frac{{V_{Zz}\left( {t->\infty} \right)} + {V_{Xx}\left( {t->\infty} \right)}}{2{V_{Yy}\left( {t->\infty} \right)}} = {\frac{1 + {\frac{3}{8}\left( {\alpha^{2} - 1} \right)}}{1 + {\frac{3}{4}\left( {\alpha^{2} - 1} \right)}}.}} & (52)\end{matrix}$

When the dip angle θ is known or estimated, the anisotropy ratio mayalternatively be determined from:

$\begin{matrix}{\frac{{V_{Xx}\left( {t->\infty} \right)} - {V_{Zz}\left( {t->\infty} \right)}}{{V_{Xx}\left( {t->\infty} \right)} + {V_{Zz}\left( {t->\infty} \right)}} = {\frac{\frac{3}{4}\left( {\alpha^{2} - 1} \right)}{2 + {\frac{3}{4}\left( {\alpha^{2} - 1} \right)}}\cos \; 2{\theta.}}} & (53)\end{matrix}$

It is further remarked that the sum of the co-axial response with the Xxcoplanar response is independent from the approach angle.

Apparent Conductivity For Co-Axial and Co-Planar Responses in aHomogeneous Anisotropic Formation

Similar to the investigation set forth above with regard to layermodels, apparent conductivity is also a useful derived formationquantity in case of an anisotropic formation layer.

The apparent conductivity is defined for both coaxial (σ_(Zz)(t)) andcoplanar (σ_(Xx)(t), σ_(Yy)(t)) responses. The apparent conductivity isthe time-varying conductivity that would give the measured coaxial orcoplanar response at time t if the formation would be homogeneous andisotropic.

As before, the time-changing apparent conductivities depend on thevoltage response in a coaxial tool (V_(zZ)(t)) or in a coplanar tool(V_(Xx)(t) at each time of measurement as:

$\begin{matrix}{{{V_{Zz}(t)} = {C\frac{\left( {\mu_{0}{\sigma_{Zz}(t)}} \right)^{3/2}}{8t^{5/2}}^{- u^{2}}}};} & (54) \\{{{V_{Xx}(t)} = {C\frac{\left( {\mu_{0}{\sigma_{Xx}(t)}} \right)^{\frac{3}{2}}}{8t^{\frac{5}{2}}}\left( {1 - u^{2}} \right)^{- u^{2}}}}{wherein}} & (55) \\{{u^{2} = {\frac{\mu_{0}{\sigma_{Zz}(t)}}{4t}L^{2}}},{or}} & \left( {56a} \right) \\{{u^{2} = {\frac{\mu_{0}{\sigma_{Xx}(t)}}{4t}L^{2}}},} & \left( {56b} \right)\end{matrix}$

Then, at large t, the apparent conductivity approaches the valuedetermined by the anisotropic conductivity and the dip angle as follows:

$\begin{matrix}{{{\sigma_{Zz}\left( t\rightarrow{large} \right)} = {\sigma_{H}\left\{ {1 + {\frac{3}{4}\left( {\alpha^{2} - 1} \right)\sin^{2}\theta}} \right\}^{\frac{2}{3}}\mspace{14mu} {for}\mspace{14mu} {coaxial}\mspace{14mu} {response}}};} & (57) \\{{{\sigma_{Xx}\left( t\rightarrow{large} \right)} = {\sigma_{H}\left\{ {1 + {\frac{3}{4}\left( {\alpha^{2} - 1} \right)\cos^{2}\theta}} \right\}^{\frac{2}{3}}\mspace{14mu} {for}\mspace{14mu} {Xx}\text{-}{coplanar}\mspace{14mu} {response}}};} & (58) \\{{\sigma_{Yy}\left( t\rightarrow{large} \right)} = {\sigma_{H}\left\{ {1 + {\frac{3}{4}\left( {\alpha^{2} - 1} \right)}} \right\}^{\frac{2}{3}}\mspace{14mu} {for}\mspace{14mu} {Yy}\text{-}{coplanar}\mspace{14mu} {{response}.}}} & (59)\end{matrix}$

In terms of the apparent conductivity,

$\begin{matrix}{{\left. {{\sigma_{Xx}(t)}^{\frac{3}{2}} + {\sigma_{Zz}(t)}^{\frac{3}{2}}} \right|_{t\rightarrow{large}} = {\sigma_{H}^{\frac{3}{2}}\left\{ {2 + {\frac{3}{4}\left( {\alpha^{2} - 1} \right)}} \right\}}};{and}} & (60) \\{\left. {{\sigma_{Xx}(t)}^{\frac{3}{2}} - {\sigma_{Zz}(t)}^{\frac{3}{2}}} \right|_{t\rightarrow{large}} = {\sigma_{H}^{\frac{3}{2}}{\left\{ {\frac{3}{4}\left( {\alpha^{2} - 1} \right)\cos \; 2\theta} \right\}.}}} & (61)\end{matrix}$

The anisotropy ratio α² may be estimated from the ratio of equations(61) and (60), and the estimated θ as:

$\begin{matrix}{\left. \frac{{\sigma_{Xx}(t)}^{3/2} - {\sigma_{Zz}(t)}^{3/2}}{{\sigma_{Xx}(t)}^{3/2} + {\sigma_{Zz}(t)}^{3/2}} \right|_{t->{large}} = {\frac{\frac{3}{4}\left( {\alpha^{2} - 1} \right)}{2 + {\frac{3}{4}\left( {\alpha^{2} - 1} \right)}}\cos \; 2{\theta.}}} & (62)\end{matrix}$

MODELING EXAMPLES

FIGS. 41 to 45 relate to transient electromagnetic inductionmeasurements, and analysis thereof, in a homogeneous anisotropicformation for various β² (in order of increasing anisotropy: 1.0; 0.8;0.6; 0.4; 0.3) for a coaxial L=1 m tool.

Of these Figures, FIG. 41 shows the calculated coaxial voltage responsesfor a formation wherein the conductivity in horizontal direction σ_(H)=1S/m (R_(H)=1 Ωm). The lines show the voltage response as a function oftime t (ranging from 1E-08 sec to 1E+00 sec on a logarithmic scale)after a step-wise sudden switching off of the transmitter. Line 101corresponds to a homogeneous isotropic formation (β²=1.0) and shouldideally correspond to a dipole solution. Lines 102, 103, 104, and 105represent increasing anisotropy and respectively correspond to β²=0.8,β²=0.6, β²=0.4, and β²=0.3.

FIG. 42 shows the apparent conductivity that has been calculated fromthe responses as shown in FIG. 41. The same line numbers have been usedas in FIG. 41.

FIG. 43 is similar to FIG. 42 but it shows the apparent conductivitythat has been derived from responses calculated for formations withσ_(H)=0.1 S/m (R_(H)=10 Ωm). The same general behavior is found.

FIG. 44 is similar to FIGS. 42 and 43, but it shows the apparentconductivity that has been derived from responses calculated forformations with σ_(H)=0.01 S/m (R_(H)=100 Ωm). The same general behavioris again found.

In each of FIGS. 42, 43, and 44, the late time apparent conductivity isconstant for each of the anisotropic factors, indicative of amacroscopically homogeneous formation. The late time apparentconductivity decreases with anisotropic factor as is expected becausethe vertical conductivity, along the principal axis of the anisotropy,is lower than the horizontal conductivity.

FIG. 45 plots the late time asymptotic value of coaxial apparentconductivity σ_(Zz)(t→∞) over σ_(H) against

$\left\{ {1 + {\frac{3}{4}\left( {\beta^{2} - 1} \right)}} \right\}^{2/3}.$

The resulting straight line demonstrates the linear relationship. Whentaking into account the anisotropy, the correct value of the horizontalformation resistivity (or conductivity) can thus be extracted from theasymptotic coaxial apparent conductivity values.

Even for highly anistropic formations, the apparent conductivity isalmost indistinguishable from apparent conductivity of a homogeneousisotropic formation with a lower conductivity. Interpretation mistakesmay thus easily be made if anisotropy is not taken into account whenanalyzing.

As follows from the above, anisotropy can be taken into account, forinstance by combining co-axial responses with coplanar responses. Theprecise embodiment depends on which of the parameters are known orestimated. The sum of the co-axial response with the Xx coplanarresponse is independent from the approach angle. If C and σ_(H) areknown or estimated then the anisotropy ratio α² follows from the latetime value of sum V_(Zz)+V_(Xx). If, on the other hand, the approachangle θ is known, C and σ_(H) don't need to be known because theanisotropy ratio α² may be derived from Eq. (53). If none of the otherparameters is known, Eq. (52) may be employed requiring combiningco-axial response with two independent co-planar responses.

Apparent Dip in a Homogeneous Anisotropic Formation

In FIG. 46, apparent dip angles θ_(app)(t) derived using Eq. (51) fromcalculated coaxial, coplanar and cross-component transient responsesfrom a L=1 m tool in a formation of R_(H)=10 Ωm and R_(V)/R_(H)=9, forvarious approach angles, or dip angles. Line 106 corresponds to θ=30°;line 107 to θ=45°; line 108 to θ=600; and line 109 to θ=75°. The dipangle is thus reflected accurately by the asymptotic value of theapparent dip. The asymptotic value is reached in approximately 1E-06sec.

Apparent Resistivity for Co-Axial and Co-Planar Responses in a FormationLayer Comprising Multiple Sub-Layers

FIG. 47 shows an electromagnetic induction tool 80 in a formation layer110 comprising a sequence or package of alternating sets of sub-layers112 and 114, set 112 having electromagnetic properties, notablyconductivity, that is different from set 114. The tool axis is depictedin the plane of the sub-layers.

While each sub-layer in the laminate of thin layers may have isotropicproperties such as isotropic conductivity, the combined effect of thesub-layers may be that the formation layer that consists of thesub-layers exhibits an anisotropic electromagnetic induction. If eachsub-layer 112, 114 in the formation layer 110 acts as an individualresistor, the macroscopic resistivity (inverse of conductivity) of theformation layer in a planar direction may be a resultant of all thelayer-resistors in parallel while the macroscopic resistivity in anormal direction (i.e. perpendicular to the layers) may be a resultantof all the layer resistors in series.

In equation form:

$\begin{matrix}{R_{V} = {\frac{1}{\Delta}{\int_{0}^{\Delta}{{R(z)} \cdot {z}}}}} & (63)\end{matrix}$

for the resistivity in the vertical, or principal direction, and

$\begin{matrix}{\sigma_{H} = {\frac{1}{\Delta}{\int_{0}^{\Delta}{{\sigma (x)} \cdot {x}}}}} & (64)\end{matrix}$

for the conductivity in the horizontal, or in-plane, directionperpendicular to the principal direction. Of course, σ_(V) can be foundusing σ_(V)=1/R_(V), and R_(H) can be found using R_(H)=1/σ_(H). Hencethe in-plane resistivity is typically lower than the resistivity in theprincipal direction. These equations also hold for more general caseswhereby the sub-layers are not of equal thickness and/or the sublayersare not of equal conductivity.

FIG. 48 shows the calculated apparent resistivity for the tool in thegeometry of FIG. 47, whereby L=1 m; the resistivity of sub-layers 112 is10 Ωm; the resistivity of sub-layers 114 is 1 Ωm, and each sub-layer is10 m of thickness. Line 115 corresponds to apparent resistivity forco-axial measurement geometry while line 116 corresponds to apparentresistivity for co-planar measurement geometry.

The apparent resistivity represented by lines 115 and 116 reflect thenear-layer resistivity of 1 Ωm at short times after the switching off ofthe transmitter. After a time span of approximately 2E-5 sec, theapparent resistivity starts to increase due to the higher resistivity of10 Ωm in the first adjacent sub-layers 112. So far, the apparentresistivity reflects what was set forth above for formations comprisingtwo or three isotropic formation layers.

However, for later times the sub-layers are no longer individuallyresolved in the responses, in which case apparent resistivity isbelieved to reflect contributions from the sub-layer where the tool 80is located, the adjacent layers and next adjacent layers, and so on.Effectively, the transient responses will show the macroscopicanisotropic behavior. In the example of FIG. 48, the collection of theisotropic sub-layers that are not individually resolved in the transientresponses are described by assuming an anisotropic layer with ananisotropic ratio of α²=R_(H)/R_(V)=1/(σ_(H)R_(V))=1/(0.55·5.5)=0.33,which can be found out using the late time apparent resistivities as setforth above for the homogeneous anisotropic formation. It may be betterto invert the responses assuming a homogenous anisotropy than to try anddetermine the individual sub-layer structure.

The dotted lines 117 and 118 in FIG. 48, which correspond to theco-axial and co-planar apparent conductivities calculated for R_(H)=1.82(i.e. 1/0.55) Ωm and R_(V)=5.5 Ωm, indeed match the drawn lines 115 and116 well, at large t.

The combined, “macroscopic,” anisotropic effect of a sub-layeredanomaly, such as is shown in FIG. 49, may also be observed. Here, theanomaly A is formed of a formation layer having a thickness Δ comprisinga thinly laminated sequence of a first formation material A1 and asecond formation material A2. FIG. 49 also indicates coaxial 60,coplanar 62, and cross-component 64 measurement arrangements, wherein atransmitter coil T and a receiver coil R are spaced a distance L apartfrom each other. The distance between the transmitter coil T and thenearest interface 55 between the near formation layer and the anomaly Ais indicated by D₁.

Using the principles set forth above, the analysis taking into accountanisotropy may be extended to multiple bedded formations, includingthose where only a distant formation layer displays macroscopicelectromagnetic induction responses (such as for instance in FIG. 49) orwhere a local formation layer wherein the transmitter and receiverantennae are located, displays anisotropic behavior but whereby one ormore other, isotropic or anisotropic layers are present at a distance.

Geosteering Applications

As stated before in this specification, electromagnetic anisotropy mayarise intrinsically in certain types of formations, such as shales. Ashale may cap a reservoir of mineral hydrocarbon fluids. It would thusbe beneficial to precisely locate a shale during drilling of a well, anddrill between for instance 10 m and 100 m below the shale to enableoptimal production of the hydrocarbon fluids from the reservoir. Thiscan be done either by traversing the shale or steering below the shalein a deviated well such as a horizontal section.

In other cases, the hydrocarbon containing reservoir may havematerialized in the form of a stack of thin sands, which itself mayexhibit anisotropic electromagnetic properties. It would be beneficialto identify the presence of such sands and steer the drilling bit intothese sands.

In each of these cases, geosteering may be accomplished by performingthe transient electromagnetic analysis while drilling and taking intoaccount formation anisotropy. This may be implemented using the systemas schematically depicted in FIG. 1A.

More generally, geosteering decisions may be taken based on locating anytype of electromagnetic anomaly using transient electromagneticresponses. Such geosteering applications allow to more accurately locatehydrocarbon fluid containing reservoirs and to more accurately drillinto such reservoirs allowing to produce hydrocarbon fluids from thereservoirs with a minimum of water.

In order to produce the mineral hydrocarbon fluid from an earthformation, a well bore may be drilled with a method comprising the stepsof:

suspending a drill string in the earth formation, the drill stringcomprising at least a drill bit and measurement sub comprising atransmitter antenna and a receiver antenna;

drilling a well bore in the earth formation;

inducing an electromagnetic field in the earth formation employing thetransmitter antenna;

detecting a transient electromagnetic response from the electromagneticfield, employing the receiver antenna;

deriving a geosteering cue from the electromagnetic response.

Drilling of the well bore may then be continued in accordance with thegeosteering cue until a reservoir containing the hydrocarbon fluid isreached.

Once the well bore extends into the reservoir containing the mineralhydrocarbon fluid, the well bore may be completed in any conventionalway and the mineral hydrocarbon fluid may be produced via the well bore.

Geosteering may be based on locating an electromagnetic anomaly in theearth formation by analysing the transient response in accordance withthe present specification, and taking a drilling decision based on thelocation relative to the measurement sub. The location of the anomalymay be expressed in terms of distance and/or direction from themeasurement sub to the anomaly.

To facilitate executing the drilling decision, the drill string maycomprise a steerable drilling system 19, as shown in FIG. 1A. Thedrilling decision may comprise controlling the direction of drilling,e.g. by utilizing the steering system 19 if provided, and/orestablishing the remaining distance to be drilled.

Accordingly, the geosteering cue may comprise information reflectingdistance between the target ahead of the bit and the bit, and/ordirection from the bit to target. Distance and direction from the bit tothe target may be calculated from the distance and direction from thetool to the bit, provided that the bit has a known location relative tothe electromagnetic measurement tool.

Transient electromagnetic induction data may be correlated with thepresence of a mineral hydrocarbon fluid containing reservoir, eitherdirectly by establishing conductivity values for the reservoir orindirectly by establishing quantitative information on formation layersthat typically surround a mineral hydrocarbon fluid containingreservoir.

In preferred embodiments, the transient electromagnetic induction data,processed in accordance with the above, is used to decide where to drillthe well bore and/or what is its preferred path or trajectory. Forinstance, one may want to stay clear from faults. Instead of that, or inaddition to that, it may be desirable to deviate from true verticaldrilling and/or to steer into the reservoir at the correct depth.

The distance from the measurement sub to an anomaly in the formation maybe determined from the time in which one of apparent conductivity andapparent resistivity begins to deviate from the corresponding one ofconductivity and resistivity of formation in which the measurement subis located and/or determining time in which one of apparent dip andapparent azimuth and cross-component response starts to deviate fromzero. The distance may also be determined from when one of apparent dipand apparent azimuth reaches an asymptotic value.

The electromagnetic anomaly may be located using at least one oftime-dependent apparent conductivity, time dependent apparentresistivity, time-dependent dip angle, and time-dependent azimuth anglefrom the time dependence of the transient response, in accordance withthe disclosure elsewhere hereinabove.

Any of the above mentioned time-dependencies can provide a usefulgeosteering cue.

Fast Imaging Utilizing Apparent Conductivity and Apparent Angle

Apparent conductivity and apparent dip may also be used to create an“image” or representation of the formation features. This isaccomplished by collecting transient apparent conductivity data atdifferent positions within the borehole.

The apparent conductivity should be constant and equal to the formationconductivity in a homogeneous formation. The deviation from a constantconductivity value at time (t) suggests the presence of a conductivityanomaly in the region specified by time (t). The collected data may beused to create an image of the formation relative to the tool.

When the apparent resistivity plots (R_(app)(z; t)) or apparentconductivity plots ((σ_(app)(z; t)) at different tool positions arearranged together to form a plot in both z- and t-coordinates, the wholeplot may be used as an image log to view the formation geometry, even ifthe layer resistivity may not be immediately accurately determined.

An example of such an image representation of the transient data asshown in FIG. 50 for a L=1 coaxial tool. The z coordinate references thetool depth along the borehole. The σ_(app)(z; t) plot shows theapproaching bed boundary as the tool moves along the borehole.

FIG. 51 shows another example. The z-coordinate represents the tooldepth along the borehole with the borehole intersecting the layerboundary in this case. The σ_(app)(z; t) plot clearly helps to visualizethe approaching and crossing the bed boundary as the tool moves alongthe borehole, for instance during drilling of the borehole.

Another example is shown in FIG. 52 wherein a 3-layer model is used inconjunction with a coaxial tool having a 1 m spacing is in two differingpositions in the formation. The results are plotted on FIG. 53A, wherethe apparent resistivity R_(app)(t) is plotted at various points as thecoaxial tool 80 approaches the resistive layer (see FIG. 53B).

FIG. 53A may be compared to FIG. 53B to discern the formation features.Starting in the 10 Ωm layer 82, the drop in R_(app)(t) is attributableto the 1 Ωm layer 83 and the subsequent increase in R_(app)(t) isattributable to the 100 Ωm layer 84. Curves (91, 92, 93) may readily befitted to the inflection points to identify the responses to the variousbeds, effectively imaging the formation. Line 91 corresponds to thedeflection points caused by the 1 Ωm bed 83, line 92 to the salt 84, andline 93 to the deflection points caused by 10 Ωm bed 82. Moreover, the 1Ωm curve may be readily attributable to direct signal pick up betweenthe transmitter and receiver when the tool is located in the 1 Ωm bed.

In still another example, the apparent dip θ_(app)(t) may be used togenerate an image log. In FIG. 54A a coaxial tool is seen as approachinga highly resistive formation at a dip angle of approximately 30 degrees.The apparent dip response is shown in FIG. 54B. As noted previously, thetime at which the apparent dip response occurs is indicative of thedistance to the formation. When the responses for different distancesare plotted together, a curve may be drawn indicative of the response asthe tool approaches the bed, as shown in FIG. 54B.

Summarising, the subterranean formation traversed by a wellbore may beimaged using a tool comprising a transmitter for transmittingelectromagnetic signals through the formation and a receiver fordetecting response signals in a procedure comprising steps wherein

the tool is brought to a first position inside the wellbore;

the transmitter is energized to propagate an electromagnetic signal intothe formation;

a response signal that has propagated through the formation is detected;

a derived quantity is calculated for the formation based on the detectedresponse signal for the formation;

the derived quantity for the formation is plotted against time.

Then the tool is moved to at least one other position within thewellbore, whereafter the steps set out above are repeated. Optionally,this can be done again. Then an image of the formation within thesubterranean formation is created based on the plots of the derivedquantity.

Optionally tool is then again moved to at least one more other positionwithin the wellbore and the whole procedure can be repeated again.

Creating the image of the formation features may include identifying oneor more inflection points on each plotted derived quantity and fitting acurve to the one or more inflection points.

Thus an image of the formation may be created using apparentconductivity/resistivity and apparent dip angle without the additionalprocessing required for inversion and extraction of information. Thisinformation is capable of providing geosteering queues as well as theability to profile subterranean formations.

1. A method of analyzing a subterranean formation traversed by awellbore, using a tool comprising a transmitter antenna and a receiverantenna, the subterranean formation comprising one or more formationlayers and the method comprising: suspending the tool inside thewellbore; inducing one or more electromagnetic fields in the formation;detecting one or more time-dependent transient response signals;analyzing the one or more time-dependent transient response signalstaking into account electromagnetic anisotropy of at least one of theformation layers.
 2. The method of claim 1, wherein the at least oneformation layer comprises three or more sub-layers.
 3. The method ofclaim 2, wherein one of the three or more sub-layers has a firstresistivity or conductivity that is different from a second resistivityor conductivity of another one of the three or more sub-layers.
 4. Themethod of claim 2, wherein the sub-layers that are not individuallyresolved in the transient response signals jointly are approximated asone anisotropic formation layer.
 5. The method of claim 1, whereinanalyzing the one or more time-dependent transient response signalstaking into account electromagnetic anisotropy includes deriving ananisotropy parameter of the at least one formation layer from thedetected one or more time-dependent transient response signals.
 6. Themethod of claim 5, wherein the anisotropy parameter comprises at leastone from a group of parameters comprising anisotropy ratio, anisotropicfactor, conductivity along a principal anisotropy axis, resistivityalong the principal anisotropy axis, conductivity in a planeperpendicular to the principal anisotropy axis, resistivity in a planeperpendicular to the principal anisotropy axis; tool axis angle relativeto the principal anisotropy axis.
 7. The method of claim 1, whereinanalyzing the one or more time-dependent transient response signalscomprises combining multi-axial transient measurements to derive ananisotropy parameter.
 8. The method of claim 1, wherein analyzing theone or more time-dependent transient response signals taking intoaccount electromagnetic anisotropy comprises deriving at least one oftime-dependent apparent conductivity, time dependent apparentresistivity, time-dependent dip angle, and time-dependent azimuth anglefrom the time dependence of the transient response signals.
 9. Themethod of claim 1, wherein one of the formation layers comprises ananomaly, and wherein analyzing the one or more time-dependent transientresponse signals comprises determining at least one of a distance and adirection between the tool and the anomaly from the one or moretime-dependent transient response signals.
 10. The method of claim 1,wherein inducing one or more electromagnetic fields in the formationcomprises generating a transmission and terminating the transmission,and detecting one or more time-dependent transient response signalscomprises measuring a receiver response as a function of time followingthe terminating the transmission.
 11. A method of producing a mineralhydrocarbon fluid from an earth formation, the method comprising stepsof: suspending a drill string in the earth formation, the drill stringcomprising at least a drill bit and measurement sub comprising atransmitter antenna and a receiver antenna; drilling a well bore in theearth formation; inducing an electromagnetic field in the earthformation employing the transmitter antenna; detecting one or moretime-dependent transient electromagnetic response signals from theelectromagnetic field, employing the receiver antenna; deriving ageosteering cue from the electromagnetic response; continue drilling thewell bore in accordance with the geosteering cue until a reservoircontaining the hydrocarbon fluid is reached; producing the hydrocarbonfluid.
 12. The method of claim 11, wherein drilling the well borecomprises operating a steerable drilling system in the earth formation.13. The method of claim 11, wherein inducing the electromagnetic fieldin the earth formation comprises generating a transmission andterminating the transmission, and detecting one or more time-dependenttransient response signals comprises measuring a receiver response as afunction of time following the terminating the transmission.
 14. Themethod of claim 11, wherein deriving the geosteering cue comprisesanalyzing the one or more transient response signals taking into accountelectromagnetic anisotropy of at least one of the formation layers. 15.The method of claim 11, wherein deriving the geosteering cue compriseslocating an electromagnetic anomaly in the earth formation based on theone or more time-dependent transient response signals.
 16. The method ofclaim 15, wherein locating the electromagnetic anomaly comprisesdetermining at least one of a distance from the measurement sub to theanomaly and a direction from the measurement sub to the anomaly.
 17. Themethod of claim 16, wherein determining the distance comprisesdetermining a time in which one of apparent conductivity and apparentresistivity begins to deviate from the corresponding one of conductivityand resistivity of formation in which the device is located.
 18. Themethod of claim 16, wherein determining the distance comprisesdetermining a time in which one of apparent dip and apparent azimuthreaches an asymptotic value.
 19. The method of claim 16, whereindetermining the distance comprises determining a time in which one ofapparent dip and apparent azimuth and cross-component response, reachesa non-zero value.
 20. The method of claim 15, wherein locating theelectromagnetic anomaly comprises deriving at least one oftime-dependent apparent conductivity, time dependent apparentresistivity, time-dependent dip angle, and time-dependent azimuth anglefrom the time dependence of the transient response.
 21. The method ofclaim 11, wherein deriving the geosteering cue comprises deriving atleast one of time-dependent apparent conductivity, time dependentapparent resistivity, time-dependent dip angle, and time-dependentazimuth angle from the time dependence of the transient response.
 22. Acomputer readable medium storing computer readable instructions thatanalyze one or more detected time-dependent transient electromagneticresponse signals that have been detected by a tool suspended inside awellbore traversing a subterranean formation after inducing one or moreelectromagnetic fields in the formation, wherein the computer readableinstructions take into account electromagnetic anisotropy of at leastone formation layer in the subterranean formation.